i^l^:..t-:'ate;:4^iu^:...M;  :^:IuiS. 


IvIBRARV 

OF  THK 

University  of  California. 

Received      dA^i^iy  .  /<^9  ?  • 

Accession  No.  ^7^  ^O      .    Class  No. 


Digitized  by  the  Internet  Arcinive 

in  2007  witii  funding  from 

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THE 


ELEMENTS 


WRITTEN     ARITHMETIC; 


COMBINING 


ANALYSIS    AND    SYNTHESIS; 


ADAPTED  TO 


THE  BEST  MODE  OF  INSTRTICTION 


FOB  BEGINNERS. 


JAMES    S.    EATON,    M.  A., 

INSTRUCTOB  IIT  PHILLIPS  ACADEMY,  ANDOVEE,  Ain>  AXTTHOB  OF  A  SERIES 
OF  ABITHMETICS. 


-^ 


BOSTON: 

29  CORNHIIiL. 

1868. 


J    J 


Entered  according  to  Act  of  Congress,  in  the  year  1868,  by 

JAMES    H.    EATON,    M.  A., 

In  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Massachusetts. 


J.  E.  Fak>vell  &  Co., 

Stereo typers  and  Printers. 

37  Congress  Street, 

Boston. 


PREFACE 


During  the  few  years  Eaton's  Series  of  Arithmetics  has 
been  before  the  Educational  Public,  it  has  been  demonstrated, 
hy  their  cordial  reception — by  their  circulation,  already  exten- 
sive and  rapidly  increasing  —  that  a  deep-felt  want  for  books  of 
their  high  character  has  been  satisfactorily  met. 

In  the  vast  field  at  the  South  now  being  opened  to  educational 
advantages  —  in  the  extensive  and  rapidly  growing  settlements  of 
the  West,  and  in  the  cities  and  manufacturing  districts  of  the 
older  States,  there  is  a  large  class  of  pupils  whose  school  days  are 
very  limited.  To  these  —  next  to  reading  and  writing  —  a  brief, 
practical  course  of  Arithmetic  must  always  form  the  most  useful  part 
of  school  training.  It  was  to  meet  the  wants  of  this  class  that  the 
present  work  was  projected  by  its  author.  Owing  to  an  unfore- 
seen event,  its  completion  has  devolved  upon  others.  This  has 
been  the  occasion  of  some  delay  in  its  publication,  which  however 
has  been  taken  advantage  of  to  make  it  as  pe'rfect  as  possible. 

This  little  work  then  is  a  short,  practical  course  of  Written 
Arithmetic,  embracing  the  topics  actually  necessary  to  be  mas- 
tered to  enable  one  to  pursue  with  intelligence  the  ordinary  busi- 
ness avocations  of  life.  Special  attention  has  been  bestowed  upon 
the  Fundamental  Rules,  United  States  Money  and  Percent- 
age, and  a  simple  but  full  exposition  of  the  New  Metric  System 
is  also  presented. 


4  PREFACE. 

In  its  preparation  no  labor  has  been  spared  to  adapt  it  to  the 
end  for  which  it  is  designed.  It  has  been  submitted  to  many  prac- 
tical teachers,  and  thus  embodies  valuable  suggestions  from  many 
sources.  Especial  credit  is  due  Mr.  J.  P.  Payson,  Master  of  the 
Grammar  School;  Chelsea,  Mass.,  and  it  makes  its  way  to  the 
public  through  the  hands  of  Mr.  James  H.  Eaton,  son  of  the 
author  of  Eaton's  Series  of  Arithmetics.  It  is  hoped  that  it  will 
prove  worthy  to  stand  beside  the  former  works  of  the  same  well- 
known  author. 

Boston,  Sept.  30,  1867. 


-i?^^ 


CONTENTS 


SIMPLE    NUMBERS. 


PACK 

Definitions 7 

Notation  and  Numeration 7 

Numeration  Table 10 

Exercises  in  Numeration 11 

Exercises  in  Notation ,  .  12 

Eoman  Notation 14 

Table  of  Koman  Numerals 14 


PAGE 

Exercises  in  Roman  Notation   ...  15 

Addition 16 

Subtraction 24 

Multiplication 35 

Division 50 

General  Principles  of  Division  ...  66 
Cancellation 69 


DENOMINATE  NUMBERS  AND  REDUCTION. 


Definitions 74 

English   Money 75 

Troy  Weight 78 

Apothecaries'  "Weight SO 

Avoirdupois  Weight 81 

Cloth  Measure 82 

Long  Measure 83 

Chain  Measure 85 


Square  Measure 86 

Solid  Measure 88 

Liquid  Measure 92 

Dry  Measure 93 

Time 94 

Circular  Measure 95 

Examples  in  Reduction 97 


GENERAL   PRINCIPLES. 

Definitions 98 1  Greatest  Common  Divisor   ....  100 

Factoring  Numbers   , 98  Least  Common  Multiple 101 

COMMON    FRACTIONS. 


Definitions   .  . ^^^ 

General  Principles 104 

Mixed  Numbers    Reduced   to  Ira- 
proper  Fractions 106 

Improper    Fractions    Reduced    to 

Whole  or  Mixed  Numbers    .  .  106 
Fractions  Reduced  to  Lowest  Terms  107 
Fraction   Multiplied   by  a    Whole 
Number 108 


Fraction  Divided  by  a  Whole  Num- 
ber     109 

Fraction  Multiplied  by  a  Fraction  .  Ill 
Fraction  Divided  by  a  Fraction  .  .  112 

Common  Denominator 113 

Addition  of  Fractions 115 

Subtraction  of  Fractions 116 

Practical  Examples 117 

Miscellaneous  Examples 118 

Analysis 119 


CONTENTS. 


DECIMAL  FRACTIONS. 


PAGE 

Definitions 123 

Numeration  Table 123 

Notation  and  Numeration 125 

Addition 126 

Subtraction 127 

Multiplication 128 


PAGK 

Division 131 

Common    Fractions    Reduced     to 

Decimals 133 

Decimals    Reduced     to     Common 

Fractions 133 

Jliscellaneous  Examples 134 


UNITED   STATES    MONEY. 


Table 136 

Definitions 136 

Practical  Examples 137 

Table  of  Aliquot  Parts 142 


Barter 144 

Bills 145 

Miscellaneous  Examples 147 


COMPOUND  NUMBERS. 


Addition 150 

Subtraction 153 

Multiplication 156 


Division .158 

Miscellaneous  Examples 159 


PERCENTAGE. 


Definitions 100 

To  find  Percentage,  the  Base  and 
Rate  being  given 162 

To  find  the  Rate,  the  Base  and  Per- 
centage being  given 163 

To  find  the  Base,  the  Rate  and  Per- 
centage being  given 164 

Interest 165 


To  find  Interest  on  any  sum  for  any 

time  at  6  per  cent 166 

To  find  Interest  at  other  Rates  than 

6  per  cent 169 

Profit  and  Loss 170 

To  find  the  Absolute  Gain  or  Loss  .  171 
To  find  Per  Cent  of  Gain  or  Loss    .  171 

To  find  the  Selling  Price 172 

To  find  the  First  Cost 173 


MISCELLANEOUS. 
Miscellaneous  Examples 1741  The  Metric  System 


177 


elemenIs^MjMetic. 


Article  1,  A  Unit  is  one,  that  is,  any  single  tiling ;  as, 
a  horse,  a  dajy  an  apple,  an  inch. 

3.  A  Number  is  a  unit  or  a  collection  of  units  ;  as,  one, 
twOf  six,  three  men,  ten  pints, 

3.  Arithmetic  is  tlic  science  of  numbers,  and  tlic  art  of 
reckoning  or  computation. 

4.  There  are  six  and  only  six  different  operations  in 
Arithmetic,  namely,  Notation,  Numeration,  Addition,  Sub. 
traction.  Multiplication,  and  Division. 


NOTATION  AND  NUxMERATION. 

S,  Notation  is  the  art  of  writing  or  expressing  numbers 
and  their  relations  to  each  other  by  means  oi  figures  and  signs. 

O.  Numeration  is  the  art  of  reading  numbers  which 
have  been  written  or  expressed  by  figures. 

T.  There  are  two  methods  of  notation  in  common  use, 
namely,  the  Arabic  and  the  Roman. 

8.  Tho  Arabic  Notation  employs  ten  figures  to  express 
numbers,  namely: 

0,        1,        2,        3,        4,       5,        6,        7,        8,        9. 

Naught,  One,     Two,    Three,    Four,    Five,      Six,     Seven,   Eight,    Nine. 

Questions.  1.  "What  is  a  Unit ?  3.  What  is  a  Number?  3.  What  is 
Arithmetic  ?  4.  How  many  operations  are  there  in  Arithmetic  ?  What  arc 
they  called  ?  5.  What  is  Notation  ?  6.  Numeration  ?  7.  How  many 
methods  of  Notation  in  common  use  ?  What  are  they  ?  8.  How  many  and 
what  figures  in  the  Arabic  Notation  ? 


8  NOTATION    AND    NUMERATION. 

9.  The  first  Arabio  figure,  0,  is  called  a  cipher ^  naugJd,  or 
zero,  and  when  used  without  any  other  figure  it  stands  for 
nothing  ;  thus,  0  apples  means  no  apples. 

Each  of  the  other- nine  figures  stands  for  or  signifies  the 
same  as  the  word  written  under  it,  and  to  distinguish  them 
from  0,  they  are  called  signifijcant  figures. 

10.  To  express  numbers  greater  than  nine,  these  figures 
arc  repeated  and  combined  in  various  ways.  Ten  is  expressed 
by  writing  the  figure  1  at  the  left  of  the  cipher;  thus,  10. 
In  like  manner  twenty,  thirty,  forty,  etc.,  are  expressed  by 
placing  2,  3,  4,  etc.,  at  the  left  of  0  ;  thus, 

20,         30,         40,        50,         60,        70,       80,        90. 

Twenty,     Tliirty,      Forty,       Fifty,        Sixty,    Seventy,  Eighty,    Ninety. 

11.  The  numbers  from  10  to  20  are  expressed  by  placing 
the  figure  1  at  the  left  of  each  significant  figure  ;  thus, 

11,  12,         13,         14,         15,         16,         17,       etc. 

Kloven,     Twelve,    Thirteen,  Fourteen,  Fifteen,  Sixteen,    Seventeen,    etc. 

In  a  similar  manner  all  the  numbers,  up  to  one  hundred, 
may  be  written  ;  thus, 

21,  36,  66,  98,  etc. 

Twenty-one,         Thirty-six,  Sixty-six,        Ninety-eight,  etc. 

12.  One  hundred  is  expressed  by  placing  the  figure  1  at 
the  left  of  two  ciphers;  thus,  100.  In  like  manner  two 
hundred,  three  hundred,  etc.,  are  written;  thus, 

200,  300,  600,  800,  etc. 

Two  hundred,    Three  hundred,      Six  hundred,      Eight  hundred,        etc. 

9.  Names  of  the  first  figure,  0?  Its  value?  What  are  the  other  figures 
called  ?  Why  ?  10.  How  are  numbers  greater  than  9  expressed  i  Illustrate. 
11.  Numbers  from  10  to  20,  how  expressed?  Other  numbers  to  100? 
la.    One  hundred,  two  hundred,  etc.,  how  expressed  ? 


NOTATION   AND    NUMERATION.  9 

13.  The  other  numbers,  up  to  one  thousand,  may  be 
expressed  by  putting  a  significant  figure  in  the  place  of  one  or 
each  of  the  ciphers  in  the  above  numbers  j  thus, 

Two  hundred  and  three,  expressed  in  figures,  is  203. 

Six  hundred  and  eighty,  •*  "  680 

Nine  hundred  and  ninety-eight,    "  "        '  998 

14.  The  simple  name  of  any  significant  figure  is  always 
the  same,  but  the  numbor  indicated  by  it  depends  upon  the 
place  the  figure  occupies ;  for  example,  6  is  always  sixy  and 
never  seven.  So  in  each  of  the  following  numbers,  2,  20,  and 
200,  the  left  hand  figure  is  twOy  but  in  the  j^rs^  it  is  two  units  ; 
in  the  second,  two  tens  or  twenty  ;  and  in  the  third,  two  hun- 
dreds. Thus  each  significant  figure  has  a  simple  or  name 
value,  and  a  local  or  place  value. 

15.  When  two  or  more  figures  are  used  together  they  are 
said  to  express  different  orders  of  units.  The  right  hand 
Jigure  represents  simple  units,  or  units  of  the  first  order  ;  the 
second  figure  represents  tens,  or  units  of  the  second  order  ;  the 
third  represents  hundreds,  or  units  of  the  third  order ;  thus, 
in  the  number  426  the  6  is  simply  six,  while  the  2  is  two  tens 
or  twenty,  and  the  4  is  four  hundreds;  and  the  number 
expressed  by  the  three  figures  taken  together  is  four  hundred 
and  twenty-six. 

16.  The  figures  of  large  numbers,  for  convenience  in 
reading,  are  often  separated  by  commas  into  groups  or 
periods  of  three  figures  each,  beginning  at  the  right.  The 
first  or  right-hand  group  contains  units,  tens,  and  hundreds, 


13.  Other  numbers  to  one  thousand  ?  14:.  Does  the  name  of  a  figure  ever 
change?  Does  its  value  change?  How  many  values  has  a  figure?  The 
names  of  those  values  ?  15.  What  is  said  of  orders  of  units  ?  16.  What 
is  said  of  groups  or  periods  of  figures  ? 


10  NOTATION   AND   NUMERATION. 

Viud  is  coMed  the  period  of  unifs ;  the  second  period  contains 
thousands,  tens  of  thousands,  and  hundreds  of  thousands,  and 
is  called  the  period  of  thousands y  etc.,  as  in  the  following 

NUMERATION  TABLE. 


02 

no 
J 

J. 

TO 
P 

O 

•^  .2 

If 

|~^ 

«+-i  ;^ 

«»H    ^ 

u^   o 

^  ^- 

reds  0 
of  Th 
sands, 

'S  § 

22^  s 

2^    1 

("Hund 
Tens 
Billio 

fHund 
Tens 
Millie 

fHund 

Tens 

[Thous 

6  9, 

5  4  0, 

7  0  6, 

4  7  6, 

5th  period, 
Trillions. 

4th  period, 

3d  period, 

2d  period, 

Billions. 

Millions. 

Thousands, 

8  4  3, 

Ist  period, 
Units. 

•IT".  The  value  of  the  figures  in  this  table,  expressed  in 
words,  is  sixty-nine  trillion,  five  hundred  and  forty  billion, 
seven  hundred  and  six  million,  four  hundred  and  seventy-six 
thousand,  eight  hundred  and  forty-three. 

Note.  The  reading  of  a  number  consists  of  two  distinct 
processes  :  First,  reading  the  order  of  the  places,  beginning  at  the 
right  hand;  thus,  units,  tens,  hundreds,  thousands,  etc.,  as  in  the 
Numeration  Table  ;  and,  second,  reading  the  value  of  the  figures, 
beginning  at  the  left,  as  in  Article  17,  above.  To  distinguish  these 
processes,  the  first  may  be  called  numerating,  and  the  second 
reading,  the  number. 

18.      The  value  of  a  figure  is  increased  tenfold  by  removing 

Name  the  periods  in  the  Numeration  Table,  beginninj?  at  the  riglit.  Name 
the  figures  in  each  group.  17.  Read  the  value  of  the  fibres  in  the  Numer- 
ation Table.  Note.  How  many  processes  in  reading  a  nuniber?  Describe 
them,  and  tell  what  they  are  called.  18.  How  is  the  value  of  a  figure 
affected  by  changing  its  place  ?    Illustrate.    What  general  law  ? 


NOTATION    AND    NUMERATION. 


11 


it  one  place  toward  the  left ;  a  hundred  fold  by  removing  it 
two  places,  etc.,  that  is,  tea  units  of  the  first  order  make  one 
ten,  ten  tens  make  one  hundred,  ten  hundreds  make  one 
thousand,  and,  in  short,  ten  units  of  any  order  make  one  unit 
of  the  next  higher  order, 

10.  The  cipher,  when  used  with  other  figures,  fills  a 
place  that  would  otherwise  be  vacant;  thus,  in  206  the  cipher 
occupies  the  place  of  tens,  because  there  are  no  tens  expressed 
in  the  given  number. 

30.  From  the  foregoing,  to  numerate  and  read  a  number 
expressed  by  figures,  we  have  the  following 

EuLE  1.  Beginning  at  the  right,  numerate,  and  point  off 
the  number  into  periods  of  three  figures  each. 

2.  Beginning  at  the  left,  read  each  period  separately,  giving 
the  name  of  each  period  except  that  of  units. 


Exercises  in  Numeration. 


SI. 

Let  the  learner  read  the  followin 

g  numbers : 

1. 

8 

11. 

4,683 

21. 

300,006 

2. 

13 

12. 

9,000 

22. 

5,634,872 

28 

13. 

35,618 

23*. 

7,402,309 

4. 

346 

14. 

40,306 

2t. 

4.040,060 

5. 

701 

15. 

75,001: 

25. 

2,008,001: 

6. 

358 

16. 

97,400 

26. 

32,468,312 

7. 

490 

17. 

66,040 

27. 

461,084,307 

8. 

8,645 

18. 

345,284 

28. 

5,329,684,119 

9. 

3,059 

19. 

549,603 

29. 

42,382,000,000 

10. 

8,006 

20. 

203,940 

30. 

702,437,600,216 

Note.  The  teacher  should  give  examples  similar  to  the  above 
upon  the  bhickboard  or  slate,  sometimes  inserting  and  sometimes 
omitting  the  commas,  until  the  pupil  can  readily  group,  numerate, 


19. 


a  numbe 


The  cipher,  for  what  used  ?    30.    Rule  for  numerating  and  reading 
i)er  ? 


12  NOTATION    AND   NUMERATION. 

and  read  all  numbers  likely  to  occur  in  his  lessons  or  general 
reading.  A  like  remark  applies  to  all  the  following  parts  of  tlie 
book.  The  teacher  should  give  many  original  examples,  varying 
in  difficulty  according  to  the  abilities  of  his  classes,  and  should 
encourage  his  pupils  to  make  examples  for  themselves  and  for  each 
other. 

33,     To  write  numbers,  we  have  this 

KuLE  1.  Beginning  at  the  left,  write  the  figures  belonging 
to  the  highest  period. 

2.  Write  the  figures  of  each  successive  period  in  their  order, 
filling  each  vacant  place  with  a  cipher. 

Exercises  in  Notation. 

33.  Let  the  learner  write  the  following  numbers  in 
figures,  and  read  them  : 

1.  Five  units  of  the    third  order   and   six   of  the   first. 

Ans.  50G. 
Note.     A  cipher  is  written  in  the  second  place,  because  no  unit 
of  the  second  order  is  given. 

2.  Three  units  of  the  fourth  order,  six  of  the  second,  and 
four  of  the  first.  '  Ans.  3,064. 

3.  Two  units  of  the  seventh  order,  one  of  the  sixth,  three  'of 
the  third,  and  five  of  the  second.  Ans.  2,100,350. 

4.  Eight  units  of  the  fifth  order,  two  of  the  third,  and  six  of 
the  first. 

5.  Six  units  of  the  eighth  order,  four  of  the  sixth,  two  of  the 
fourth,  and  five  of  the  third. 

6.  Nine  units  of  the  sixth  order,  two  of  the  fourth,  and 
eight  of  the  first. 

33.    Rule  for  writing  a  number.    Note.    In  Notation  where  should  0  be 
written .' 


NOTATION   AND   NUMERATION.  13 

Y.  What  orders  of  units  are  there  in  the  number  3,462,895  ? 
How  many  units  in  each  order  ? 

8.  What  orders  of  units  in  the  number  62,304,500  ?  'How 
many  units  in  each  order  ? 

9.  How  many  tens  in  46  ?  How  many  units  beside  the 
tens  ?     How  many  units  in  the  whole  of  the  number  ? 

10.  In  347  how  many  hundreds?  How  many  tens  in  the 
tens'  place  ?  How  many  units  in  the  units'  place  ?  How  many 
tens  in  the  number  ?     How  many  units  in  the  number  ? 

24l,     Write  the  following  numbers  in  figures : 

1.  Two  hundred  and  fifty-six.  Ans.  256. 

2.  Fifty.four.  Ans.  54. 

3.  Six  thousand  and  nineteen.  Ans.  6,019.     * 

4.  One  thousand  eight  hundred  and  sixty-five. 

5.  Four  hundred  and  forty. 

6.  Twenty-five  thousand  two  hundred  and  forty-nine. 

7.  Two  hundred  and  forty-five  thousand  six  hundred  and 
fifty-four. 

8.  Five  million  six  hundred  thousand  eight  hundred  and 
sixteen. 

9.  Twenty-two  million  two  hundred  and  twenty-two  thou- 
sand two  hundred  and  twenty-two. 

10.  Five  hundred  and  six  million  forty  thousand  two  hun- 
dred and  four. 

11.  Four  billion  eight  million  six  thousand  eight  hundred 
and  ten. 

1 2.  Thirty-five  trillion  four  hundred  and  six  billion  eight 
hundred  and  twenty  million  two  hundred  and  eighteen  thou- 
sand four  hundred  and  sixty- seven. 


14 


NOTATION   AND   NUMEKATION. 


35.  The  Ko MAN  Notation  employs  seven  capital  letters 
to  express  numbers,  viz. : 

I,    *  V,       X,        L,  C,  D,  M. 

One,      Five,      Ton,     Fifty,      One  hundred.      Five  hundred,    One  thousand. 

All  other  numbers  may  be  expressed  by  combining  and 
repeating  these  letters. 

36.  The  Koman  Notation  is  based  on  the  following  prin- 
ciples : 

1st.  When  two  or  more  letters  of  equal  value  are  united,  or 
when  one  of  less  value  follows  one  of  greater,  the  sum  of  the 
values  is  indicated  ;  thus,  XX  stands  for  20,  XXX  for  30, 
LXV  for  65,  DC  for  600,  MDCCLXVIII  for  1768. 

2d.  When  a  letter  of  less  value  is  placed  before  one  of 
4 greater,  the  difference  of  their  values  is  indicated ;  thus,  IV 
stands  for  4,  IX  for  9,  XL  for  40,  XO  for  90. 

3d.  When  a  letter  of  less  value  stands  between  two  of  greater 
value,  the  less  is  to  bo  taken  from  the  sum  of  the  other  two  ; 
thus,  XIV  stands  for  14,  XIX  for  19,  CXL  for  140. 


TABLE  OF  KOMAN  NUMERALS. 


I 

1 

X 

10 

XIX 

II 

2 

XI 

11 

XX 

III 

3 

XII 

12 

XXI 

IV 

4 

XIII 

13 

XXII 

V 

5 

XIV 

14 

XXIV 

VI 

6 

XV 

15 

XXV 

VII 

7 

XVI 

16 

XXIX 

VIII 

8 

XVII 

17 

XXX 

IX 

9 

XVIII 

18 

XL 

19 
20 
21 
22 
24 
25 
29 
30 
40 


85.    How  many  and  what  characters  are  employed  in  the  Roman  Nota- 
tion?   Value  of  each?    How  may  other  numbers  be  expressed? 
ae.   What  is  the  first  principle  in  the  Ilomau  Notation  ?    Second  ?    Third  ? 


NOTATION   AND    NUMERATION. 


15 


L 

50 

DC 

600 

MDCCXLIX 

1749 

LX 

CO 

DCCCG 

000 

MDCCCXVI 

1816 

XG 

90 

M 

1000 

MDCCCXLI 

1841 

C 

100 

MD 

1500 

MDCCCXLIX 

1849 

COCO 

400 

MDO 

1600 

MDCCCLVII 

1857 

D 

500 

MDCLXV  1665 

MDCCCLXVI 

1866 

Exercises  in  Koman  Notation. 

S7.     Express  the  following  numbers  by  letters: 

1.  Nine.  Ans.  IX. 

2.  Fifteen.  '  Ans.  XV. 

3.  Eighteen. 

4.  Twenty-four. 

5.  Twenty-six. 

6.  Thirty-nine. 

7.  Forty. 

8.  Sixty. 

9.  One  hundred  and  eighty-four. 

1 0.  One  hundred  and  ninety-six. 

11.  One  thousand  six  hundred  and  forty-six. 

1 2.  The  present  year,  A.  D. . 

38.  Besides  the  Arabic  and  Roman  figures,  there  are 
various  marks  used  to  indicate  certain  relations  between  num- 
bers and  operations  to  be  performed  on  them,  as,  for  example, 
the  sign  of  equality,  ^=. ;  the  sign  of  addition,  -|-  ;  the  sign  oj 
subtraction,  — ;  etc. 

These  signs  will  be  given,  and  their  uses  explained 
hereafter,  when  their  aid  is  needed. 


28.    What  characters  are  used  in  Arithmetic  besides  the  Arabic  and  Koman 
figures?    For  what? 


16 


ADDITION. 


ADDITION. 

39,  Three  apples  and  four  apples  are  how  many  apples  ? 
Ans.      Three  apples  and  four  apples  are  seven  apples. 

This  is  a  question  in  addition. 

30.  Addition  is  the  process  of  finding  how  many  units 
there  are  in  two  or  more  numbers  of  the  same  kind  taken  to- 
gether.    The  result  of  the  addition  is  called  the  sum  or  amount. 

ADDITION  TABLE. 


2  and  1 

are  3 

3  and  1  are  4 

4  and 

1 

are  5 

5 

and  1  arc  6 

2    "    2 

"    4 

3    "     2  "    5 

4    " 

2 

"     6 

5 

«'    2   "    7 

2    "    3 

"    5 

3    "     3  "     6 

4    " 

3 

"     7 

5 

«*    3   "    8 

2    "    4 

"     6 

3    "     4  "     7 

4    " 

4 

"     8 

5 

"    4   '*    9 

2    ♦'    5 

"    7 

3    "     5  "     8 

4    " 

5 

"     9 

5 

"    5   "  10 

2    "    6 

"    8 

3    "     6  "     9 

4    " 

6 

"  10 

5 

"    6   "  11 

2    *'    7 

"    9 

3    "     7  "  10 

4    " 

7 

"  11 

5 

"    7   "  12 

2    "    8 

"  10 

3    "     8  "  11 

4    " 

8 

"  12 

5 

"    8   "  13 

2    "    9 

"  11 

3    "     9  "  12 

4    " 

9 

"   13 

5 

"    9   "  14 

2    "10 

"  12 

3    "  10  "   13 

4    " 

10 

"  14 

5 

"10   "  15 

6  and  1 

are  7 

7  and  1  are  8 

8  and 

1 

are  9 

9  and  1  are  10 

6    "    2 

"    8 

7    "    2  ♦♦     9 

8    " 

2 

"  10 

9 

"    2    "  11 

6    "    3 

"    9 

7    "    3  "    10 

8    " 

3 

"  11 

9 

"    3    "  12 

6    "    4 

"  10 

7    "    4  "   11 

8    " 

4 

"  12 

9 

"    4    "  13 

6    "    5 

"  11 

7    "    5  "   12 

8    " 

5 

"  13 

9 

"    5    "  14 

6    '*    6 

"  12 

7    "    6  "   13 

8    " 

6 

"  14 

9 

"    6    "  15 

6    '•    7 

"  13 

7    "    7  "    14 

8    " 

7 

"  15 

9 

"    7    "  16 

6    "    8 

"  14 

7    ^'    8  "    15 

8    " 

8 

"  16 

9 

"    8    "  17 

6    "    9 

•'  15 

7    "    9  "    16 

8    ^' 

9 

"  17 

9 

'•    9    "  18 

6    "10 

"  16 

7    "10  "   17 

8    " 

10 

"  18 

9 

"10    "  19 

30.    What  is  Addition  f    What  is  tlie  reswZ^  called  ? 


ADDITION.  17 


Mental  Exescises, 


Ex.  1.  Robert  has  5  cents  in  one  hand,  and  3  cents  in  the 
other  ;  how  many  cents  has  he  in  both  hands  ?  Ans.  8. 

2.  John  bought  a  pencil  for  6  cents,  and  some  paper  for  5 
cents  ;  how  many  cents  did  he  pay  for  both  ? 

3.  Greorge  has  7  chickens  and  David  has  8  ;  how  many  have 
both?  Ans.  15. 

4.  Mary  has  6  tulips  and  9  roses;  how  many  blossoms  has 
she? 

5.  Daniel  caught  9  fishes,  Abel  caught  6,  and  James  caught 
5  ;  how  many  did  they  all  catch  ? 

6.  A  farmer  had  6  cows  in  one  pasture,  8  in  another,  and  7 
in  another ;  how  many  cows  had  he  in  the  three  pastures  ? 

7.  I  paid  9  dollars  for  a  barrel  of  flour,  8  dollars  for  a  box 
of  sugar,  and  5  dollars  for  a  cheese ;  how  many  dollars  did  I 
pay  for  all?  Ans.  22. 

8.  A  man  bought  a  ton  of  coal  for  8  dollars,  a  cord  of 
wood  for  6  dollars,  and  a  stove  for  9  dollars ;  what  did  he  pay 
for  all? 

9.  Charles  has  5  marbles,  Albert  has  7,  and  Edward  has  9  ; 
how  many  have  they  all  ? 

10.  A  farmer  has  8  sheep  in  one  pen,  9  in  another,  and  as 
many  in  a  third  pen  as  in  both  the  others ;  how  many  has  he 
in  the  third  pen  ?  Ans.  1 7. 

11.  A  gardener  raised  3  bushels  of  cherries,  2  bushels  of 
currants,  5  bushels  of  peaches,  and  8  bushels  of  pears;  how 
many  bushels  of  fruit  did  he  raise  ? 

12.  George  paid  10  cents  for  a  writing-book,  8  cents  for  a 
pen-holder,  2  cents  for  pens,  and  6  cents  for  ink ;  how  much 
did  he  pay  for  all  ? 


18 


ADDITION. 


31.  A  Sign  is  a  mark  which  indicates  an  operation  to  be 
performed,  or  which  is  used  to  shorten  some  expression. 

33.  This  mark,  $,  is  often  used  as  a  sign  of  the  word 
dollar  or  dollars  ;  thus,  %\  stands  for  one  dollar,  $6  stands  for 
six  dollars. 

Note.  It  is  customary  to  separate  dollars  and  cents  by  a  period; 
thus,  f  4.25  stands  for  four  dollars  and  twenty-five  cents. 

33.  The  sign  of  equality,  =,  signifies  that  the  quantities 
between  which  it  stands  are  equal  to  each  other ;  thus,  ^  I  ;= 
100  cents;  that  is,  one  dollar  equals  one  hundred  cents. 

34.  The  sign  of  addition,  -(-,  called  j»^m5  or  and,  denotes 
that  the  quantities  between  which  it  stands  are  to  be  added  to- 
gether ;  thus,  3  +  2  =  5;  that  is,  three  plus  two  equal  five, 
or,  three  and  two  are  five. 

Ex.  12.     How  many  are  3  +  6  +  4?     Ans,  12. 

13.     How  many  are  2  +  6  +  5?     3  +  8  +  4? 
.      14.     How  many  are  5  +  3  +  6?     9  +  2  +  6? 

15.  How  many  are  8  +  6  +  5?     9+3  +  7? 

16.  How  many  are  7  +  9  +  4?     6  +  9  +  8? 
3^.     Let  the  pupil  frequently  review  the  following 

Exercises  in  Addition. 


No.  1. 

No.  2. 

No.  3. 

No.  4. 

No.  5. 

4  +  3 

6  +  5 

2-1-8 

7  +  3 

5  +  7 

2  +  6 

5  +  5 

7  +  9 

6  +  8 

6  +  9 

7  +  3 

6+4 

1  +  8 

3  +  7 

9+3 

8  +  1 

8  +  2 

3  +  6 

8  +  0 

2  +  5 

10  +  3 

3  +  5 

8+1 

4  +  5 

5  +  8 

5  +  6 

4  +  9 

9  +  6 

7  +  6 

5  +  6 

7  +  5 

2  +  6 

4  +  7 

9  +  8 

9+2 

31.  What  is  a  sig^n?  3a.  Make  the  sign  of  dollars  on  the  black-board. 
How  are  dollars  and  cents  separated  ?    Give  an  example.    Another. 

33.  Make  the  sign  of  equality.  What  does  it  mean  ?  Illustrate.  34.  Make' 
the  sign  of  addition.    What  is  it  called .-'    What  does  it  mean  ? 


ADDITION, 


19 


"       No.  6. 

No.  7. 

No.  8. 

No.  9. 

No.  10.     ( 

9  +  4 

1+4 

10+    3 

2+    1 

H 

h    9 

6--7 

7  +  8 

8  +  10 

6  +  10 

0- 

-   '8 

8  +  9 

•  8  +  5 

3+    8 

5  +  11 

10- 

-    ^ 

G  --2 

2  +  0 

2+7 

3  +  12 

6  - 

-11 

1  --  5 

5  +  4 

9+    7 

9+0 

1- 

-    2 

2  --  2 

3  +  1 

1+    3 

8+    8 

10- 

-    7 

5  +  1 

2  +  4 

6+    1 

9+9 

7H 

hio 

No.  11, 

No.  12. 

No.  13. 

No.  14. 

No.  15. 

8+    6 

3+9 

3+    4 

4+    8 

6+3 

10+5 

7+    7 

7+    2 

3+2 

7  --  12 

11+2 

6+6 

12+5 

3+3 

12—    4 

G  +  12 

10+9 

10+6 

12+8 

11  --  11 

10+8 

12+6 

11+    7 

10  +  11 

5  --  12 

9+    1 

9  +  10 

9  +  12 

12+7 

11  --    9 

7  +  11 

8-1-12 

3  +  11 

11+8 

12+9 

Written   Exercises. 
36.     To  add  when  the  numbers  are  large,  and  the 
amount  of  each  column  is  less  than  ten. 

Ex.  1.  A  farmer  sold  234  bushels  of  corn,  423  bushels  of 
oats,  and  141  bushels  of  wheat;  how  many  bushels  of  grain 
did  he  sell  ? 

Having  for  convenience  arranged  the  numbers 

OPERATION,     so  that  uuits  stand  under  units,  tens  under  tens, 

234         etc.,  add  the  units ;  thus,  1  and  3  are  4,  and  4 

423         are  8,  and  set  the  8  under  the  column  of  units. 

141         Then,  add  the  tens;  thus,  4  and  2  are  6,  and  3 

are  9,  and  set  the  9  under  the  column  of  tens, 

Sum  798         and   so  proceed  till  all  the  columns  are  added. 
Thus  we  find  that  the  entire  sum  is  7  hundreds,  9  tens,  and 
8  units,  or  t98  bushels  the  answer. 


36.    How  are  numbers  arranged  for  addition  ? 
added  first  ?    "What  is  done  with  the  Bum  ? 


Why  ?    Which  column  is 


20  ADDITION. 

In  like  manner  add  the  numbers  in  the  following  examples : 

Ex.  2.         3.  4.  5.  6.  7.  8. 

$1.90   242   143   $26.01   324   1240  51234 

2.47   126   421    12.31    23   2036   2130 

3.11   211   235    41.32   241   3712    513 


Ans.  $7. 

48 

799 

6988 

9. 

10. 

11. 

12. 

13. 

14. 

15. 

Miles. 

Bushels. 

Men. 

Apples. 

Sheep. 

Birds. 

Days. 

1310 

3241 

4120 

4160 

203 

1321 

3122 

3247 

1302 

312 

1306 

6120 

3200 

2231 

2131 

2144 

2103 

2012 

62 

2134 

2101 

6687  6655 

16.  In  1850  the  population  of  Virginia  was  1,421,661,  and 
that  of  Vermont  was  314,120  ;  what  was  the  total  population 
of  Virginia  and  Vermont  in  1850  ?  Ans.  1,735,781. 

17.  In  1860  the  population  of  Massachusetts  was  1,231,065 
and  that  of  Kentucky  was  1,155,713  ;  what  was  the  total  popu- 
lation of  Massachusetts  and  Kentucky  in  1860? 

Ans.  2,386,778. 

18.  A  gentleman  paid  $135  for  a  horse,  $243  for  a  chaise, 
and  $121  for  a  harness ;  what  did  he  pay  for  all  ? 

19.  Add  2316,  3120,  1201,  and  2002.  Ans.  8669. 

20.  Add  $35.41,  $21.24,  $1.32,  and  $2.01.  Ans.  $59.98. 

21.  Add  43216,  20431,  14030.  Ans.  77,677. 

22.  AVhat  is  the  sum  of  3241  +  2312  +  1203  +  3120? 
28.     What  is  the  sum  of  1325  -f  2312  -f  1321  +  4031  ? 

24.  What  is  the  sum  of  1242  +  2123  +  1312  +  2112? 

Ans.  6789. 

25.  What  is  the  sum  of  3124  +  1232  +  2113  +  1220? 

26.  What  is  the  sum  of  23102  +  52454  +  24342  ? 

27.  What  is  the  sum  of  15323  +  32354  +  41302  ? 


ADDITION.  21 

37,  To  add  when  the  amount  of  any  column  Is  ten 
or  more. 

28.  A  farmer  raised  473  bushels  of  potatoes,  285  bushels 
of  onions,  568  bushels  of  carrots,  and  359  bushels  of  turnips; 
how  many  bushels  of  vegetables  did  he  raise  ?      Ans.  1685. 

Having  arranged  the  numbers  so  that  units 
OPERATION,  stand  under  units,  tens  under  tens,  etc.,  as 
473  .  in  example  1,  add  the  numbers  m  the  column 
285  of  units;  thus,  9  and  8  are  17,  and  5  arc 
668  22,  and  3  are  25  units,  (=  2  tens  and  5 
359         units).     The  5  units  are  set  under  the  column 

of  units  and  the  2  tens  are  added  to  the  tens 

Ans.  1685  given  in  the  example;  thus,  2  and  5  are  7 
and  6  are  13,  and  8  are  21,  and  7  are  28  tens 
(=  2  hundreds  and  8  tens).  The  8  tens  are  set  under  the 
tens,  and  the  2  hundreds  are  added  to  the  hundreds  in  the 
example,  giving  16  hundreds,  or  1  thousand  and  6  hundreds, 
which,  written  in  their  proper  places,  give  1685  for  the 
answer. 

38.  In  the  same  manner,  add  the  numbers  in  the  follow- 
ing short  columns f  and  also  add  across  the  page,  as  suggested 
by  the  signs. 

29.  3846  +  2843  +  63542  +  35842  +  91326  +  73241 

30.  8305      3654        82735        12600        82145        38642 

31.  9160      5003        230G4        81264        34208        26341 


21311     lin05  129706  138224 

32.  3462  -f  1538  +  56421  +  36245  +  35496  +  82437 

33.  1354      6242        91367        24687        23549        43621 

34.  1534      6215        13579        21683        35462        10820 

35.  5104      3160       20013        61000       301Q4r^    28006 

11454  181380  1246U 


37.    Explain  the  operation  in  Ex.  28., 


Hiirm^ 


22 


ADDITION. 


36.  4006+3567  +  41323  +  30000  +  5436 

37.  5143    2G4     346    3812      46 

38.  5274   3S0G    5148     346     876 

39.  8463     88   63405   87420   45362 


+    284 

3864 

29 

389 


22886  110227 

40.  8716  +    501  +   432167   + 

41.  4822  9             9S721 

42.  1920  2001                 702 

43.  1861  92                   96 


51720 

67958  +  8957351 
2780  2761852 

8765  8578127 

83217  101 


531686 


20297431 


30.  In  solving  the  foregoing  examples,  the  learner  has 
already  become  familiar  with  all  the  operations  in  addition; 
but  to  enable  him  readily  to  tell  how  to  add,  we  give  the 
following 

KuLE.  Write  the  numbers  in  order,  units  under  units,  tens 
under  tens,  etc.  Draw  a  line  beneath,  add  together  the  figures 
in  the  units  column,  and  if  the  sum  be  less  than  ten  set  it  und^r 
the  column  ;  but  if  the  sum  bs  teii  or  more,  write  the  units  as 
before,  and  add  the  tens  to  the  next  column.  Thus  proceed 
till  all  the  columns  are  added. 

40*  Proof.  The  usual  mode  of  proof  is  to  begin  at  the 
top  and  add  downward.  If  the  work  is  right,  the  two  sums 
will  be  alike. 

Note  1.  By  this  process,  we  combine  the  figures  differently,  and 
hence  shall  probably  detect  any  mistake  which  may  have  been  made 
in  adding  upward. 


39.  Why  is  a  Rule  for  addition  given  ?  Repeat  the  Rule.  If  the  amount  of 
any  column  is  10  or  more,  where  is  the  right-hand  figure  of  the  amount  written  i 
Why .''    What  is  done  with  the  left-hand  figure  or  figures .-'    Why  ? 

40.  How  is  addition  proved?  Why  not  a,M  upward  a  second  timet  In 
addition  is  it  desirable  to  name  the  figures  as  we  add  themf    Why  not? 


ILLUSTRATION. 


ADDITION .  23 

Ex.  44.    In  adding  upward  we  say  4  and  6  are  10,  and  5 

are  15,  and  8  are  23,  etc. ;  but  in  adding 

doxmiwardy  we  say  8  and  5  are  13,  and  6 

53468         are  19,  and  4  are  23,  etc.  ;  thus  obtaining 

72635         the  same  residt,  but  by  different  comhina- 

24376         tions.- 

27594 
If  we  do  not  obtain  the  same  result  by 

Sum,  178073         the  two  methods,  one  operation  or  the  other 

is  wrong,  perhaps  both,  and  the  work  must 

Iroof,  178073         \fQ  carefully  performed  again. 

Note.  In  adding  it  is  not  usually  desirable  to  name  the  figures 
tliat  we  add ;  thus,  in  Ex.  44,  instead  of  saying  4  and  6  are  10,  and 
5  are  15,  and  8  are  23,  it  is  shorter  and  therefore  better  to  say,  4, 10, 
15,  23;  and  then  setting  down  the  3,  say  2,  11,  18,  21,  27,  etc.     . 

45.  A  grain  dealer  bought  3756  bushels  of  wheat  of  A, 
2347  bushels  of  B,  1346  bushels  of  C,  and  5468  bushels  of 
D ;  how  many  bushels  of  wheat  did  he  buy?      Ans.  12917. 

46.  I  paid  8  3465  for  a  farm,  S15000  for  a  mill,  $  6795 
for  a  lot  of  wool,  and  $  4620  for  40  shares  of  railroad  stock  ; 
how  much  did  I  pay  for  all  this  property?      Ans.  $29880. 

47.  Bought  3  city  lots  for  $15345,  and  sold  them  so  as 
to  gain  $  3639  ;  what  sum  did  I  receive  for  them? 

Ans.  $18984. 

48.  A  man  commenced  trade  with  $  5345,  and  in  one  year 
he  gained  $  3462 ;  what  was  he  worth  at  the  end  of  the 
year  ? 

49.  Add  three  hundred  and  twenty. five ;  two  thousand 
one  hundred  and  fifty- four ;  two  hundred  and  fourteen ;  twenty- 
three  thousand  five  hundred  and  forty-one  ;  and  three  hundred 
and  seventy-five.  Ans.  26609. 

60.     What  is  the   sum  of  thirty-four  thousand  five  hun- 


24  SUBTRACTION. 

dred  and  forty-six  ;  five  million,  two  hundred  and  seventy-six 
thousand,  four  hundred  and  nineteen ;  and  forty-two  million , 
six  hundred  and  twenty-four  thousand,  five  hundred  and  eighty 
seven?  Ans.  47,935,552. 

51.  England  and  Wales  contain  about  55,100  square 
miles  ;  Scotland,  29,G00  ;  and  Ireland,  32,000 ;  what  is  the 
area  of  the  British  Islands  ? 

52.  The  population  of  England  in  1851  was  10,921,888  ; 
of  Scotland,  2,888,742  ;  of  Wales,  1,005,721 ;  and  of  Ireland, 
6,515,794;  what  was  the  population  of  Great  Britain  and 
Ireland?  Ans.  27,332,145. 

53.  In  1850  the  population  of  New  York  was  515,547; 
of  Philadelphia,  340,045;  of  Baltimore,  169,054  ;  of  Boston, 
136,881;  of  New  Orleans,  116,375;  and  of  Cincinnati,  115, 
436  ;  what  was  the  number  of  inhabitants  in  these  six  cities  in 
1850?  Ans.  1,393,338. 


SUBTKACTION. 

4:1.  Three  apples  taken  from  seven  apples  leave  how 
many  apples  ?  Ans.  Three  apples  from  seven  apples  leave 
four  apples.     This  is  a  question  in  Subtraction. 

4^.  Subtraction  is  taking  a  less  number  from  a  greater 
number  of  the  same  kind,  to  find  their  difference,. 

The  greater  number  is  called  the  minuend  ;  the  less  number, 
the  SUBTRAHEND  ;  and  the  difference,  the  eemainder. 

4a.    WhatisSubtxactiou.'  What  the  Minuend .'   Subtrahend?  Remainder? 


SUBTRACTION, 


25 


SUBTRACTION   TABLE. 


1  from  2  leaves  1 

2  from  3  leaves  1 

3  from  4  leaves  1 

1      "     3      "      2 

2       "     4       * 

*      2 

3      ^ 

'     5       "      2 

1       "     4      ''      3 

2       "     5       * 

'      3 

3      • 

'     6      "      3 

1       '^     5      "      4 

2      -     6       ' 

'      4 

3      ' 

'     7      "      4 

1       "     6      "      5 

2       '^     7'    ' 

'      5 

3      ' 

'     8      '•      5 

1      "     7      "      6 

2      "     8       ' 

'      C 

3      ' 

'     9      "      6 

1      "     8      *,'      7 

2-9       ' 

*      7 

3      * 

'  10      "     *7 

1      **     9      *•      8 

2       "10      ' 

^      8 

3      ' 

Ml       "8 

1       "  10      "      9 

2      "  11       ' 

'      9 

3       ' 

'  12      "      9 

1       «  11      *♦    10 

2       "12      ' 

'    1^ 

3       ' 

'  13      "    10 

4  from  5  leaves  1 

5  from  6  leaves  1 

6  from  7  leaves  1 

4      "     6      "      2 

0       "     7       ' 

'      2 

6      ' 

'     8      "      2 

4      "     7      "      3 

5       "     8       ' 

'      3 

6       « 

'     9      "      3 

4      '«     8      "      4 

5       "     9       ' 

'      4 

6      ' 

'  10      "      4 

4      "     9      "      5 

5       "10 

'      5 

6       « 

'  11       "      5 

4      ♦*  10      ♦*      6 

5       ♦*  11       ' 

'      G 

6       ' 

'  12      "      6 

4      "  11      "      7 

5       "12      ♦ 

*      7 

6       ♦ 

'  13      "      7 

4      "  12      "      8 

5       "  13       ♦ 

♦      8 

6       ' 

*  14      "      8 

4      "  13       «      9 

5       "14 

'      9 

6       ' 

*  15       "      9 

4       "  14      ♦•    10 

5-     "  15       ' 

*    10 

6       ' 

'  16       "    10 

7  from  8  leaves  1 

8  from  9  lea 

ves  1 

9  from  10  leaves  1 

7      "     9      "      2 

8      "10      ' 

'      2 

9       ' 

'   11       "      2 

7       "  10      '*      3 

8       "11       ' 

'      3 

9       ' 

M2       "      3 

7      "  11      "      4 

8       "12       ' 

'      4 

9       ' 

'13       "4 

7       "  12      "      5 

8       "13       ' 

'      5 

9       ' 

'  14      "      5 

7      "  13      "      6 

8       "14      ' 

'      G 

9       ' 

'  15       "      6 

7      "  14      '*      7 

8      "15      ' 

*      7 

9       ' 

'  16       "      7 

7      "  15      "      8 

8       "  IG      * 

'      8 

9       • 

*  17       "      8. 

7      ^'  IG      •'      9 

8       "  17      ' 

'      9 

9       ' 

'  18       "      9 

7      "  17      "    10 

8       "18      * 

'    10 

9       ' 

•  19       "    10 

26  SUBTRACTION^. 

Mental  Exercises. 
Ex.  1.  Joseph  has  8  marbles  in  his  right  hand,  and  5  in  his 
left  hand  ;  how  many  more  marbles  has  he  in  his  right  hand 
than  in  his  left?  Ans.  3. 

2.  Thomas  paid  10  cents  for  a  melon,  and  4  cents  for  an 
orange ;  how  much  more  did  the  melon  cost  than  the  orange  ? 

3.  Daniel  paid  $  1 2  for  a  colt  and  $  5  for  a  lamb ;  how  much 
leas  did  the  lamb  cost  than  the  colt  ? 

4.  A  boy  having   15  peaches  gave  away  8  of  them ;  how 
many  had  he  remaining? 

5.  A  man  owing  $  17  paid  $  9  ;  how  much  did  he  then  owe? 

6.  Bought  goods  for  $9  and  sold  them  for  $.13  ;  how  much 
did  I  gain  ?  Ans.  $  4. 

7.  Sold  goods  for  $15,  which  was  $6  more  than  they  cost 
me  ;  what  did  I  pay  for  them  ? 

8.  William  is  18  years  old  and  George  is  9  years  younger ; 
how  old  is  George? 

9.  John  had  1 7  cents  and  spent  9  of  them  ;  how  many  cents 
had  he  then  ? 

10.  A  tailor  had  15  yards  of  cloth,  from  which  he  sold  9 
yards  ;  how  many  yards  remained  ?  Ans.  6. 

11.  Samuel  is  16  years  old  and  David  is  9  ;  how  much  older 
is  Samuel  than  David? 

12.  Isaac  had  12  marbles,  but  has  lost  7  of  them;  how 
many  marbles  has  he  now  ? 

4:3.  The  sign  of  subtraction,  — ,  called  minus  or  less, 
signifies  that  the  number  after  it  is  to  be  taken  from  the  num- 
ber before  it ;  thus,  7  —  4=3;  that  is,  seven  minus  four,  or 
seven  less  four,  equals  three. 

Ex.  13.  How  many  are  9  —  5?  Ans.  4. 

43.  Make  the  sign  of  Subtraction  on  the  black-board.  What  is  it  called  ? 
What  docs  it  mean  i    Illustrate. 


SUBTRACTION, 


27 


14.  How  many  are    8  —  6?     12  —  3?     10  —  7? 

15.  How  many  are  12  — 5?       9  —  6?     11  —  5? 

16.  How  many  are  16  — 7?     15  —  9?     13  —  8? 

17.  How  many  arc  17  — 6?     12  —  8?     18  —  9? 

18.  How  many  are  18  — 7?     16—9?     14—9? 
44:.     Lef:  the  pupil  frequently  review  the  following 

Exercises  in  Subtraction. 


No.  1. 

No.  2. 

No.  3.  1 

No.  4. 

No.  5. 

6  —  2 

9  —  5 

7  —  4 

8  —  6 

13—6 

8  —  5 

6  —  3 

9  —  6 

7  —  7 

6—4 

.  3  —  1 

10  —  4 

12  —  5 

7-0 

8—  3 

9  —  7 

7  —  5 

8  —  7 

9  —  3 

10—5 

7  —  3 

3  —  3 

4—1 

2  —  2 

7—  1 

5  —  4 

3  —  0 

5  —  3 

6  —  5 

11—3 

4  —  2 

7  —  2 

7  —  6 

15  —  9 

12—4 

2—  1 

8  —  4 

9  —  8 

12  —  8 

15  —  10 

No.  6. 

No.  7. 

No.  8. 

No.  9. 

No.  10. 

9  —  4 

10  —  6 

12—  9 

15—7 

14—8 

8  —  2 

12  —  3 

14—  6 

17—9 

16—6 

12  —  6 

16  —  4 

11—7 

14—5 

10—9 

10  —  7 

11  —  5 

4—  3 

10—8 

13—4 

12  —  7 

18  —  2 

9—  2 

9—  1 

15—5 

14  —  2 

14  —  7 

11—  6 

10—0 

18—  7 

16  —  7 

8  —  1 

12—  10 

10—10 

17—8 

15  —  8 

11  —  4 

15  — -6 

18—8 

14—4 

No.  IL 

No.  12. 

No.  13. 

No.  14. 

No.  15. 

17—  6 

15  —  11 

18—  6 

17  —  11 

16  —  12 

16—  9 

17  —  10 

16—  10 

18—5 

18—  4 

14—  10 

14—9 

14—12 

16—8 

17—5 

18—  9 

11—8 

12—  1 

13—9 

14—3 

12—11 

16—5 

15—4 

15—3 

17—  7 

15  —  12 

10—3 

13—8 

18  —  16 

16  —  12 

13—7 

13—5 

18—12 

16—2 

13  —  11 

18—10 

18—11 

15  —  13 

14  —  11 

11—  9 

28 


SUBTRACTION, 


Writtejj    Exercises. 
45.     To  subtract  when  no  figure  in  the  subtrahend 
is  fjreater  than  the  fio^ure  above  it. 


Ex.  1.  From  837  take  523. 


Ins.  314. 


Operation. 

Minuend,       837 
Subtrahend,  523 

Kemainder,  314 


Having  written  the  less  number  under 
the  greater,  units  under  units,  tens  under 
tens,  etc.,  we  say  3  from  7  leaves  4,  2 
from  3  leaves  1,  and  5  from  8  leaves  3  ; 
therefore  the  remainder  is  314. 
In  like  manner  solve  the  following  examples  : 
Ex.  2.  3.  4.  5. 


Erom  $53.68 
Take    $21.43 


$  736.45 
$325.13 


38697 
13543 


Ans.  $32.25       $411.32 

7.  8. 

Hours.  Men. 

Erom  9368  65439 

Take  3215  25316 


Women. 

63548 
21410 


386495 
243345 

143150 
10. 

Children. 
390642 

180321 


6. 

836942' 
314241 


11. 

Horses. 

897436 
135223 


40123  210321 

12.  By  the  census  of  1860,  there  were  326072  inhabitants  in 
New  Hampshire,  and  628276  in  Maine;  how  much  less  was 
the  population  of  New  Hampshire  than  of  Maine  ? 

13.  By  the  census  of  1860,  the  population  of  Mississippi  was 
791396,  and  that  of  the  United  States  Territories  was  220143  ; 
how  many  more  people  were  there  in  Mississippi  than  in  the 
Territories  ?  Ans.  5  7 1 25  3. 

14.  A  farmer  bought  a  farm  for  $  3465,  and  sold  it  for 
$ 4689  ;  how  much  did  he  gain?  Ans.  $  1224. 

15.  How  many  are  29  less  16  ?     876  less  346  ? 


45.    How  are  numbers  arranged  for  Subtraction  r"    Why?    Which  figure  is 
subtracted  first  ?    Where  is  the  Kemainder  written  ? 


SUBTRACTION.  29 

16.  How  many  are  89  less    74  ?      963  less  241  ? 

17.  Howmany  are  836— -215?     8360—6320? 

18.  Howmanyare  869—349?     9386  —  2150? 

40.  To  subtract  when  any  figure  in  the  subtrahend 
is  greater  than  the  fio^ure  above  it. 

19.  From  863  take  249.  Ans.  614. 

Two  methods  for  explaining  this'opera- 
Operation.  ^Jqjj  ^j.g  jj^  common  use. 

Minuend,       863  1st.    As  we  cannot  take  9  units  from  3 

Subtrahend,  249        units,  07ie  of  the  6  tens  is  put  with  the  3 

^       .    ,  units,  making  13  units,  and  then,  9  units 

Remainder,    614        ^        ,o       -1.1  a       -j.       i.-  v  •        t. 

from  13  units  leave  4  units,  which  is  set 

under  the  units.      Now,  as  one  of  the  6  ietis  has  been  used, 

only  5  tens  remain  in  the  minuend,  and  4  tens  from  5  tens 

leave  1  ten,  and,  finally,  2  hundreds  from   8  hundreds  leave 

6  hundreds;  therefore  the  entire  remainder  is  614. 

2d.  We  may  add  10  units  (equal  to  I  ten)  to  the  three 
units,  making  13  unit?.  From  this  sum  we  subtract  the  9 
units.  In  subtracting  the  next  column,  instead  of  taking  away 
1  of  the  6  tens  in  the  minuend,  we  may  add  1  ten  to  the  4 
tens  in  the  subtrahend,  and  then  take  the  sum  (5  tens)  from 
the  6  tens,  and  the  result  is  1  ten  as  by  the  former  process. 

The  second  mode  depends  on  the  principle,  that  if  two  num- 
bers are  equally  increased,  the  difference  between  them  remains 
unchanged.  Now,  in  solving  Ex.  19  by  the  second  method, 
we  add  10  units  to  the  minuend,  and  1  ten  (the  same  as  10 
units)  to  the  subtrahend^  and  therefore  find  the  same  remainder 
as  by  the  first  method. 

46.  How  many  methods  of  subtracting  when  a  figure  of  the  subtrahend 
is  greater  than  the  figure  over  it '  Explain  the  first  method.  Explain  the 
second.  The  second  depends  on  what  principle .''  Is  the  same  number  added 
to  mimtend  and  subtrahend  ?   Mow  1 


30 


SUBTRACTION. 


47.  In  the  same  manner  solve  the  following  examples, 
taking  each  lower  number  from  the  one  over  it  in  each  exam- 
ple ;  also,  subtract  in  the  manner  indicated  by  the  signs. 


20. 


21. 


22. 


23. 


("86326  —  43710     53684—36146     74668 


i3462 


|34613  —  23620     21392  —  19324     43158  —  25319 


61713 

f 58327 


32292   16822         28143 
36-118  66888  —  43682  83621  —  34261 


\43618  —  18294  51364  —  35176  52S42  —  21638 


18124 

(73926  —  53614     83654 
■[61498—39182       8263 


8506 

46839  93654 
3642   8432 


12623 

9342 

584 


(99594 


14432 

74660 


46832 


(81940—50706  36481 


43197 

39481 
22814 


8758 

73162  —  68243 
61928  —  34821 


23954  11234 

48.  The  pupil  having  become  familiar  with  the  modes  of 
subtracting,  we  aid  him  by  giving  the  following : 

EuLE  1.  Write  the  less  number  under  the  greater,  units 
under  units,  tens  under  tens,  etc.,  and  draw  a  line  beneath. 

2.  Beginning  at  the  right  hand,  take  each  figure  in  the  sub- 
trahend from  the  figure  above  it,  and  set  tJie  remai7ider  under 
the  line. 

3.  Jf  any  figure  in  the  subtrahend  is  greater  than  the 
figure  above  it,  add  ten  to  the  upper  figure  and  take  the  lower 
figure  from  the  sum  ;  set  down  the  remainder,  and  considering 
the  next  figure  in  the  minuend  one  less,  or  the  next  figure  in 
the  subtrahend  one  greater,  proceed  as  before. 


48.    What  is  the  Rule  for  Subtraction  ? 


SUBTRACTION. 


31 


49;  Proof.  Add  the  subtrahend  and  the  remainder 
together,  and  the  sum  should  be  the  minuend. 

Note  1.  This  proof  rests  on  the  self-evident  truth,  that  the 
whole  of  a  thing  is  equal  to  the  sum  of  all  its  parts  ;  thus,  the  min- 
tiendis  separated  into  the  two  parts,  subtrahend  and  remainder; 
hence  the  sum  of  tliose  parts  must  be  the  minuend. 


ILLUSTRATION. 

Minuend,      8264: 
Subtrahend,  3692 

Eemainder,  4572 


Ex.  24.  As  the  sum  of  the  subtrahend 
and  remainder  is  the  minuend,  the  work 
is  supposed  to  be  right. 


Proof,  8264 

25. 

Inches. 

From   8365 
Take    4879 


26. 

Men. 

636554 

482732 


Rem.,  3486 


27. 

Gallons. 
96G482 
3S1779 

584703 


28. 

Apples. 
835670 
482984 


•       Proof,  8365 
(7)  (9)  (13) 
From    8     0      3 
Take    2     6       7 


Ans. 


29.  Here,  we  cannot  take  7  from  3, 
nor  can  we  borrow  from  the  tens'  place, 
as  that  place  is  occupied  by  0 ;  but  we 
can  borrow  one  of  the  8  hundreds,  and 
separate  the  one  hundred  into  9  tens  and 
10  units  ;  then,  putting  the  9  tens  in  the  place  of  tens,  and 
adding  the  10  units  to  the  3  units  in  the  minuend,  we  can 
subtract  7  from  13,  6  from  9,  and  2  from  7. 

Note  2.  This  process  will  probably  be  more  readily  understood 
by  the  young  learner  than  the  second  method  given  in  the  rule, 
tliougli  the  latter,  for  convenience,  is  usually  adopted. 


49.    What  the  proof?     On  what  principle  does  the  proof  rest  f    Illustrate. 
Explain  Ex.  29.    Which  mode  of  subtracting  is  more  readily  understood.'' 
Which  more  convenient  ? 


32 


SUliiK 

AUTIUJX. 

30. 

31. 

32. 

33. 

34. 

From  $5304: 

Days. 
6403 

Sheep. 

6030 

Miles. 

9084 

Bushels, 

8005 

Take    $2457 

3846 

2684 

7692 

3689 

$2847  2346  4316    • 

35.  Washington  was  born  in  1732,  and  died  in  1799;  at 
what  age  did  he  die  ?  Ans.  67, 

36.  How  many  years  have  passed  since  the  discovery  of 
America  in  1492  ? 

37.  Jamestown,  in  Virginia,  was  settled  in  1607;  how 
many  years  from  that  date  was  the  Declaration  of  Independence 
in  1776  ?  Ans.  169. 

38.  Queen  Victoria  was  born  in  1819  ;  how  old  was  she  in 
1865?  Ans.  46. 

39.  A  merchant  bought  goods  for  $  3846,  and  sold  the  same 
for  $  5050  ;  what  was  his  gain  ?  Ans.  $  1204. 

40.  A  merchant  paid  $  8004  for  goods,  and  sold  the  same 
for  $  684G  ;  what  was  his  loss? 

41.  How  many  years  from  the  discovery  of  America  by 
Columbus,  in  1492,  to  the  settlement  of  Plymouth  by  the 
Puritans,  in  1620?  Ans.  128. 

42.  In  1864  a  man  died  at  the  age  of  87  years  ;  in  what 
year  was  he  born  ? 

43.  The  sum  of  two  numbers  is  80304,  and  the  greater 
number  is  54836  ;  what  is  the  less?  Ans.  25468. 

44.  The  less  of  two  numbers  is  34685,  and  their  sum  is 
90304  ;  what  is  the  greater?  Ans.  55619. 

45.  The  difference  between  two  numbers  is  3684,  and  the 
greater  number  is  8002  ;  what  is  the  less  ? 

46.  From  one  thousand  eight  hundred  and  sixty-fivo 
one  thousand  four  hundred  and  ninety-two. 


SUBTRACTION.  33 

47.  From  two  million,  three  hundred  and  sixty-one  thou- 
sand, four  huudred  and  seventeen,  take  one  million,  five  hun- 
dred and  forty- six  thousand,  two  hundred  and  eighty-nine. 

Ans.  815128. 

48.  Suppose  the  distance  from  the  earth  to  the  sun  is 
94879956  miles,  and  that  from  the  earth  to  the  moon  is 
240000  miles ;  how  much  farther  is  the  sun  than  the  moon 
from  the  earth  ? 

49.  The  population  of  the  United  States  was  31,443,790 
in  1860,  and  23,191,876  in  1850 ;  what  was  the  increase  in 
ten  years?  Ans.  8,251,914. 

60.  Suppose  the  outstanding  public  debt  of  the  United 
States  to  be  $  2,800,000,000,  and  that  $  125,375,287  now 
in  the  treasury  be  applied  to  its  payment,  what  would  then  be 
their  indebtedness  ? 


EXAMPLES  IN  ADDITION  AND   SUBTRACTION. 

1.  From  the  sum  of  94  and  86,  take  117.  Ans.  63. 

2.  From  the  sum  of  the  three  numbers,  629,  493,  and  896, 
take  the  sum  of  968  and  563.  Ans.  487. 

3.  I  owe  three  notes,  whose  sum  is  $  3895  ;  one  of  these 
notes  is  for  S  1348,  another  for  $  863  ;  for  how  much  is  the 
third?  Ans.  $1684. 

4.  A  farmer  having  1275  acres  of  land,  sold  318  acres  at 
one  time,  227  at  another,  and  175  at  another ;  how  many  acres 
has  he  remaining  ? 

5.  If  a  man's  income  is  $1865  a  year,  and  he  pays  $200  for 
rent,  $468  for  food,  $278  for  clothing,  and  $712  for  other 
expenses,  how  much  will  he  save  in  the  year  ?     Ans.  $  207. 

7.  How  many  are  876  +  392  +  847  —  963  ? 

8.  How  many  are  986  +  389  -f-  549  —  846  ? 


34  SUBTRACTION. 

9.  Two  men  start  from  the  same  place  and  travel  in  the 
same  direction,  one  goes  125  miles,  the  other  876,  how  far 
apart  are  they  ?  How  far  if  they  had  travelled  in  opposite 
directions  ? 

10.  From  T,000,000  subtract  8901  +  101. 

Ans.  6990998. 

11.  A  man  purchased  a  farm  for  $  6890,  and  having  paid 
$  575  for  an  additional  piece  of  land,  he  sells  the  whole  for 
$  7500 ;  does  he  gain  or  lose,  and  how  much? 

12.  In  a  Union  school  there  are  four  departments  ;  in  the 
first  there  are  125  scholars,  in  the  second  379,  in  the  third 
437,  and  in  the  fourth  487  ;  how  many  scholars  does  it  con- 
tain ?    If  692  are  boys,  how  many  are  girls? 

First  Ans.  1428.     Second  Ans.  736. 

13.  A  general  started  oat  on  a  campaign  with  three  regi- 
ments of  soldiers,  the  first  numbered  1025  men,  the  second 
975,  the  third  875  ;  after  a  battle  he  finds  but  2575  in  all 
reported  fit  for  duty ;  how  many  men  has  he  lost  ? 

14.  How  many  are  687  +  594  +  369  —  918? 

15.  In  1850,  the  population  of  New  York  was  615547; 
that  of  Philadelphia,  340045;  of  Baltimore,  169054;  and  of 
Boston,  136881.  At  the  same  time,  the  population  of  London 
was  about  2363241 ;  what  was  the  difference  between  the 
population  of  London  and  the  aggregate  population  of  the  four 
cities  named  in  the  United  States  ?  Ans.  1201714. 

16.  A  merchant  bought  some  flour  for  $  347,  some  rye  for 
$  236,  and  some  oats  for  $  563  ;  he  sold  the  whole  for  $  1275. 
Did  he  gain  or  lose  ?     How  much  ? 

17.  Mr.  Jones  gives  $  2376.43  to  his  four  sons,  as  follows : 
to  Daniel,  $  534.68  ;  to  James,  $  354.68  ;  to  Thomas,  $  486.39  ; 
and  the  rest  to  David.     What  does  David  receive? 

Ans.  81000.68. 


MULTirLICATIOX.  35 


MULTIPLICATION. 

50.  In  1  bushel  there  are  32  quarts  ;  how  many  quarts 
are  there  in  8  bushels? 

1st  Method,  2d  Method,           This  example  may  be  solved 

BY  ADDITION.  BY  MULTIPLICATION,     by  addition,  as  by  the  1st 

3  2  3  2         method ;    but  as  there  will 

3  2  8         evidently  be  8  times  as  many 

3  2  quarts  in  8  bushels  as  there 

3  2  Product,  2  5  6         are  in  1  bushel,  it  may  be 

3  2  more  briefly  solved  as  by  the 

3  2  2d  method ;  thus,  8  times  2  units  are  1 6  units, 

3  2  =  1  ten  and  6  units ;  write  the  6  units  in  the 

3  2  place  of  units,  and  then  say  8  times  3  tens  are 

— I —  24  tens,  which,  increased  by  the  1  ten  previ- 

Sum,   2  5  6  Q^giy  obtained,  make  25  tens,  =  2  hundreds 

and  5  tens,  and  when  these  are  written  in  their  proper  places 

we  have  256  quarts  for  the  true  result.     This,  when  solved 

by  the  2d  method,  is  a  question  in  multiplication. 

51.  Multiplication  is  a  short  method  of  adding  equal 
numbers;  or,  it  is  a  short  method  of  finding  how  many 
units  there  are  in  any  number  of  times  a  given  number. 

The  number  repeated  is  called  the  multiplicand  ;  the  num- 
her  showing  how  many  times  the  multiplicand  is  taken  is  the 
multiplier  ;  the  sum,  or  result  of  the  multiplication,  is  the 
product.  The  Multiplicand  and  Multiplier  are  called  Fac- 
tors. 

50.  Explain  the  two  methods  of  solving  the  above  example.  "Which  is 
best? 

51.  What  is  Multiplication  ?  Another  definition.  What  is  the  Multipli- 
cand? Multiplier?  Product?  Factors?  53.  Repeat  the  Multiplication 
Table. 


36 


MULTIPLICATION. 


53.      The  pupil,  before  advancing  further,  should  learn 


the  following 


MULTIPLICATION   TABLE. 


Once 

1  is  1 

2  "  2 
3 
4 
5 
6 
7 


Twice 

1 

are  2] 

2 

<( 

4 

3 

(( 

6 

4 

(( 

8 

5 

<« 

10 

6 

(( 

12 

7 

(< 

14 

8 

(( 

16 

9 

<« 

18 

10 

<< 

20 

11 

<( 

22 

12 

•' 

2\: 

Three  times 

1 

are  3 

2 

<( 

6 

3 

<( 

9 

4 

(« 

12 

5 

<i 

15 

6 

<( 

18 

7 

(( 

21 

8 

<( 

24 

9 

<< 

27 

10 

iC 

30 

11 

K 

33 

12 

(( 

36 

Four  times 
1  are  4 


2 

3 
4 
6 
6 
7 
8 
9 

10 
11 
12 


8 
12 

16 
20 
24 
28 
32 
36 
40 
44 
48 


Five  times 
1  are  5 


2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 


10 
15 
20 
25 
SO 
35 
40 
45 
50 
55 
60 


Six  times 
1  are  6 


2 

3 
4 
5 
6 
7 
8 
9 

10 
11 
12 


12 

18 
24 
30 
36 
42 
48 
54 
60 
QQ 
72 


Seven  times 

Eight  times 

Nine  times 

1  are  7 

1  are  8 

1  are  9 

2  "  14 

2  "  16 

2  "    18 

3  "21 

3  "  24 

3  "    27 

4  "28 

4  "  32 

4  "   36 

5  "  35 

5  "  40 

5  "   45 

6   "42 

6  "  48 

6  "    54 

7  "  49 

7  "  56 

7  "    63 

8  "  56 

8  "  64 

8  "   72 

9  "  63 

9  "  72 

9  "    81 

10  "  70 

10  "  80 

10  "    90 

11   "  77 

11   "  88 

11  "    99 

12  "  84 

12  "  96 

12  "  108 

Ten  times 
1  are  10 
20 


10 
11 
12 


30 

40 

50 

60 

70 

80 

90 

100 

110 

120 


Eleven  times 

1 

ire  11 

2 

«    22 

3 

'    33 

4 

.    44 

5 

'    55 

6 

'    QQ 

7 

c       rjrj 

8 

'    88 

9 

'    99 

10 

"110 

11 

'121 

12 

♦132 

Twelve  times 
1  are  12 


2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 


24 

36 

48 

60 

72 

84 

96 

108 

120 

132 

144 


MULTIPLICATION, 


Mental  Exercises. 


1.  How  many  leaves  will  7  sheets  of  paper  make  if  each 
sheet  is  folded  in  8  leaves?  Ans.  56. 

2.  In  how  many  days  can  I  do  as  much  work  as  9  men  can 
do  in  5  days  ? 

8.  One  peck  contains  8  quarts ;  how  many  quarts  are  there 
in  3  pecks  ? 

4.  If  9  yards  of  cloth  are  required  to  make  1  garment,  how 
many  yards  are  required  to  make  8  such  garments? 

5.  How  many  men  can  do  as  much  work  in  1  day  as  6  men 
can  do  in  9  days?  Ans.  54. 

6.  If  you  solve  9  examples  each  hour,  how  many  examples 
will  you  solve  in  9  hours  ? 

7.  When  flour  is  worth  $  10  per  barrel,  how  much  must 
be  paid  for  7  barrels? 

8.  If  12  inches  make  1  foot,  how  many  inches  are  there 
in  3  feet?  Ans.  36. 

9.  In  1  year  there  are  12  months;  how  many  months  are 
there  in  2  years  ?  4  years  ?  3  years  ?  7  years  ?  5  years  ? 
8  years  ? 

10.  If  I  deposit  $10  a  month  in  a  savings  bank,  how 
much  shall  I  deposit  in  5  months?  In  4  months?  In  8 
months  ?     In  9  months  ? 

^3.  The  sign  of  multiplication,  X»  signifies  that  the  two 
numbers  between  which  it  stands  are  to  be  multiplied  together  ; 
thus,  6  X  5  rr:  30 ;  that  is,  six  multiplied  by  five  equals  thirty,  or 
six  times  five  are  thirty. 

Ex.     1 1.  How  many  are  7  X  4  ?  Ans.  28. 


53.    Make  the  sign  of  multiplication  on  the  black-board.    What  docs  it 
signify? 


38 


MULTIPLICATION. 


54:.     Review  until  familiar  the  following 

Exercises  in   Multiplication. 


No.  1. 

No.  2. 

No.  3. 

No.  4. 

No.  5. 

3X5 

4  X  6 

8X4 

8X7 

6  X  3 

6  X  4 

7  X  3 

7  X  9 

6  X  6 

7  X  5 

5X8 

6  X  8 

5X6 

5X0 

9X5 

4X7 

5  X  4 

4  X  9 

5  X  1 

8  X  6 

7  X  8 

8  X  3 

6  X  5 

3  X  7 

7  X  4 

9  X  2 

9  X  4 

7  X  6 

4X8 

6X7 

2X8 

6  X  9 

5  X  7 

7X2 

2  X  9 

1  X  6 

3X8 

8X8 

9  X  3 

3X6 

No,  G. 

No.  t. 

No.  8. 

No.  9. 

No.    10. 

7X7 

10  X     3 

1  X     8 

12  X     8 

11  X     1 

8X5 

8X2 

T  X  10 

1  X  11 

12  X     6 

10  X  2 

6  X     1 

12  X    5 
8  X  10 
7X12 

9  X    8 

9  X  10 

3  X  9 

t  X  10 

4  X  10 

10  X     6 

5  X  5 

6X2 

8  X     1 

4  X  12 

8  X  9 

12  X     3 

12  X     4 

12  X     2 

4X4 

4X5 

4X11 

8  X  11 

9  X     9 

6  X  11 

9  X  6 

5  X  12 

9  X     1 

11  X    3 

8  X  12 

No.    11. 

No.    12. 

No.    13. 

No.    14. 

No.    15. 

12  X     1 

11  X     6 

12  X  11 

10  X  10 

11  X     9 

10  X     8 

10  X     7 

11  X  12 

11  X  11 

5X3 

C  X  12 

5X11 

3  X  11 

9  X  12 

11  X     2 

6X0 

4X3 

5X9 

3  X  10 

10  X  11 

6  X     1 

5  X  10 

9  X  11 

2  X  12 

10  X  12 

10  X     9 

12  X     1 

3  X  12 

11  X    8 

12  X  10 

12  X     9 

10  X     5 

11  X     5 

1  X     1 

4X2 

11  X    4 

12  X  10 

10  X    4 

2  X  11 

12  X  12 

MULTIPLICATION.  39 

Written  Exercises. 
55.     To  multiply  by  a  single  figure. 
Ex.  1.  Multiply  879  by  6  Ans.  52H. 

FIRST  OPERATION. 

879 
6 

Since  879  is  to  be  multiplied  by  6,  77      .. 

L      1       i..,        .             ,,.,.*  54  units, 

each  order  of  its  units  must  be  multiplied  a  n  2. 

1.    ^     T-         11           1                       .       „  *  ^  tens, 

by  b  ;  hence  the  product  must  consist  of  4  «  h      1    rl 

5i  units,  42  tens,  and  48   hundreds;  

and,  therefore,  the  product  is  5  2  7  4 

Instead  of  writing   the  products  of  the   units,  tens,  etc., 

separately  and  adding  the  several  partial  products,  it  is  more 

convenient,  and,  therefore,  customary  to  multiply  as  in  the 

SEcoxD  OPERATION.        Here,  by  the  same  plan  as  in  Article 

8  7  9     ,        60,  we  say,  6  times  9  are  54  ;  set  down 

Q  the  4,  and  then  say,  6  times  7  are  42, 

and  5  are  47  ;  set  down  the  7,  and  then 

5  2  7  4  say,  G  times  8  are  48  and  4  are  52 ;  set 

the  2  and  5  in  their  proper  places,  and 

,  the  entire  product,  is  the  same  as  before. 

56.     From  the  above  we  have  the  following 

Rule.      Write   the   multiplier  under  the  muJtiplicand,  and 

draw  a  line  beneath ;  multiply  the  units  of  the  multiplicand^ 

set  the  units  of  the  product  under  the  multiplier,  and  add  the 

teiis,  if  any,  to  the  product  of  the  tens,  and  so  proceed  till 

the  example  is  solved. 

55.  Explain  the  First  Solution  of  Example  one.  Explain  the  Second  So- 
lution. Are  the  two  methods  alike  in  principle  ?  VThich  is  the  more  convenient 
in  practice  ?  56.  Which  figure  of  the  multiplicand  is  multiplied  first  ?  Where 
are  the  units  of  the  product  written  ?  What  is  done  with  the  tens  ?  Repeat  the 
Rule. 


±\J 

iU  U  1j  riii-ljlUAl  iUJN  . 

Ex.  2. 

3. 

4. 

Multiplicand, 

$5.3  6             6  42 

954 

Multiplier, 

7                  4 

8 

Product, 

$3  7.5  2          25  6  8 

5. 

6. 

7. 

9843 

3645 

634508 

2 

9 

3 

19686 

8. 

9. 

10. 

439684 

389642 

823465 

5 

7 

6 

2198420 

4940790 

11. 

12. 

13. 

854839 

$  9  0  8  6.4  7 

856732 

9 

9 

4 

8  1  7  7  8.2  3 

14.  Multiply  63842  by  5. 

Ans.  319210. 

15.  Multiply  30687  by  7. 

16.  Multiply  86054  by  6. 

Ans.  516324. 

17.  Multiply  12309  by  4. 

18.  Multiply  59480  by  8. 

Ans.  475840. 

19.  Multiply  40069  by  9. 

20.  Multiply  562347  by  2. 

21.  Multiply  385462  by  3. 

2  2.  How  many  ; 

are  3584  X  7? 

Ans.  25088. 

23.  How  many 

are  8639  X  4? 

24.  If  1  farm 

is  worth  $6375,  what 

are  6  farms  worth,  at 

the  same  price  ? 

MULTIPLICATION. 


41 


25.  In  1   square  foot  there  are   144  square  inches;  how 
many  square  inches  are  there  in  9  square  feet  ? 

26.  In  1  solid  foot  there  are  1728  solid  inches;  how  many 
solid  inches  are  there  in  8  solid  feet?  Ans.  13824. 

57,     To  multiply  by  a  number  expressed  by  two  (fr 
more  figures. 

27.  In  1  bushel  there  are  32  quarts;  how  many  quarts  are 
there  in  48  bushels  ? 


OPERATION. 


Multiplicand, 
Multiplier, 

Partial  "j 

Products,      f 


32 

48 

256 
128 


First  multiply  by  8,  as  though 
8  were  the  only  figure  in  the 
multiplier  ;  then  multiply  by  4, 
and  set  the  first  figure  of  this 
product  in  the  place  of  te7is ;  for 
multiplying  by  the  4  tens  is  the 
same  as  multiplying  by  40,  and 
=  8  tens;  that  is,  the  pro- 
Having  multiplied  by  each  figure 


True  Product,     15  3  6 
40  times  2  units  are  80 
duct  of  tcnits  by  tens  is  tens. 
in  the  multiplier,  the  sum  of  the  partial  products  is  the  true 
product. 

•>8.     Similar  reasoning  applies  however  many  figures  there 
may  be  in  the  multiplier.     Hence, 

EuLE  1.     Set   the   multiplier  under  the  multiplicand,  and 
draw  a  line  beneath. 

2.  Beginning  at  the  right  hand  of  the  multiplicand,  multiply 
the  midtiplicand  by  each  Jig  ure  of  themidtiplier,  setting  tJie  first 

figure  of  each  partial  product  directly  under  the  figure  of  the 
multiplier  which  produces  it. 

3.  The  Sum  of  the  partial  products  willbe  the  true  product. 


57,    Explain  the  solution  of  Example  30.    Where  is  the  first  figure  of  each 
partial  product  written  ?   "Why?    58.   Kepeat  the  JRule. 


42  MULTIPLICATION. 

50,     Proof.     Multiply  the  multiplier  hy  the  multiplicand, 
and,  if  correct,  the  result  will  be  like  the  first  product. 

Note.     This  proof  rests  on  the  principle,  that  tlie  order  of  tlie 
factors  is  immaterial ;  3X4=4X3;  6X2X7=2X7X5=2X5X7,  etc. 

'  Ex.  28.     Multiply  669  by  418. 

OPERATION.  PROOF. 

MultiplicanJ,  5  69  47  8 

Multiplier,  4  7  8  ,     *        5  6  9 


4552  4302 

3983  2868 

2276  2390 


Product,  271982  =  271982 

60.  In  the  same  manner  solve  the  following  examples, 
multiplying  each  upper  number  by  the  one  under  it  in  each 
example ;  also  multiply  in  the  manner  indicated  by  the  signs. 

oq(8X9      24  X  n      634X376     4362X3264 
^•|6X7      21X14      428X245     2468X1357 

g^f7X6      36X24      568X492     5486X3698 


4^ 


X8      27X32      634X346     2534x4368 

Q,    (9X9      46X54      648X396     8682X3842 
"^^'18X6      35X43      827X423     6342x4362 

„^f9X5      88X77      986X684    9999X6843 
"^"'18X7      64X72      379X793     4682X7953 

33.  If  a  steamboat  sails  12  miles  per  hour,  how  far  will  she 
sail  in  24  hours  ;  that  is  in  1  day? 

34.  If  a  steamboat  sails  288  miles  per  day,  how  far  will 
she  sail  in  28  days  ? 

59.    Proof?    Principle?' 


MULTIPLICATION.  43 

« 
;7  acres  of  land,  at  $  13.36  per 
acre?  Ans.  $494.32. 

36.  How  long  will  it  take  1  man  to  do  as  much  work  as 
24  men  can  do  in  75  days?  Ans.  1800  days. 

37.  How  far  will  a  horse  travel  in  27  days,  if  he  travels  37 
miles  per  day  ? 

38.  How  many  yards  of  cloth  in  33  pieces,  if  each  piece 
contains  54  yards?  Ans.   1782. 

39.  Multiply  two  hundred  and  fifteen  thousand  eight  hun- 
dred and  forty-seven  by  six  hundred  and  fifty-nine. 

Ans.  142,243,173. 

40.  "What  is  the  cost  of  building  243  miles  of  railroad,  at 
$48,750  per  mile? 

41.  if  a  garrison  of  soldiers  eat  5876  pounds  of  bread  per 
day,  how  much  will  they  eat  in  365  days? 

01.  Ciphers  between  the  significant  figures  of  the  multi- 
plier may  be  neglected,  taking  care  to  set  the  first  figure  of  each 
partial  product  directly  under  the  figure  of  the  multiplier 
which  gives  that  product. 

Ex.  42.     Multiply  7543  by  2005. 

This  is  only  carrying  out  the  principle 

OPERATION.  ^.^   addition)    of    setting    units   under 

7543     units,  tens  under  tens,  etc.    The  2  of  the 

2005     multiplier  is  2000,  and  2000  times  3  is 

"TZTTT     6000,    therefore    the    6    of   the  partial 

15086  product  should  be  written  in  the  place  of 

thousands  ;  that  is,  directly  under  the  2 

Product,  15123715     of  the  multiplier. 

43.     Multiply  3642  by  3008.  Ans.  10955136. 


61.    What  may  be  done  with  ciphers  between  the  significant  figures  of  the 
multiplier  ?    What  care  is  required  ?    Principle  ? 


44  MULTIPLIOATIOIT. 

* 

44.  What  cost  507  miles  of  railroad  at  $  3G4-8  per  mile? 

Ans.  $1,849,536. 

45.  How  many  lemons  in  806  boxes,  if  each  box  contains 
309  lemons? 

46.  How  many  pounds  of  cotton  in  3004  bales,  each  bale 
containing  537  pounds?  Ans.  1,613.148. 

Contractions. 

G2,  The  rules  already  given  are  sufficient  for  all  examples 
that  can  arise  in  multiplication,  but  there  are  various  devices 
for  shortening  the  process  in  particular  cases. 

63.     To  multiply  by  a  composite  number. 

A  Composite  Xumber  is  i\iQ  product  of  two  or  more  numbers; 
15  is  a  composite  number,  whose  factors  are  3  and  5  ;  12  is  a 
composite  number,  whose  factors  are  2  and  6,  or  3  and  4,  or 
2,  2,  and  3.  It  will  be  observed  that  a  composite  number  may 
hmie  several  sets  of  factors. 

47.     How  many  dollars  have  35  men,  if  each  man  has  $  43  ? 

The  35  men  may  be  sepa- 
rated into  7  groups  of  5  men 
each.  Now  1  group  of  5  men 
will  have  5  times  S  43,  := 
$215,  and  if  1  group  has 
$215,  evidently  7  groups 
will  have  7  times  $  215  z= 
$1505,  Ans.  That  is,  7 
times  5  times  a  number  are 
Product,  $  1505     35  times  that  number. 


63.  What  ifl  said  of  the  rules  already  given  for  Multiplication  ?  What  of 
shorter  modes  ? 

63.  What  is  a  composite  number  ?  May  a  composite  number  have  more 
than  one  set  of  factors  ?    Rule  for  multiplying  by  a  composite  number  ? 


OPERATION. 

35  r=  5  X  7. 

Multiplicand, 

1st  Factor  of  Multiplier, 

$43 
5 

2d  Factor  of  Multiplier, 

$215 

7 

MULTIPLICATION.  45 

48.     Multiply  367  by  168.  Ans.  61656. 

FIRST  OPERA TIOX.  SE0OND    OPERATION. 

168  =  8  X  7  X  3.  168  =  4  X  7  X  6. 

Multiplicand,  3  6  7  3  6  7 

1st  Factor  of  Multiplier,  8  4 


2936  1468 

2(i  Factor  of  Multiplier,  7  7 


*     20552  10276 

8d  Factor  of  Multiplier,  3  6 


Product,  61656  =  61656 

Several  other  sets  of  factors  of  168  may  be  used,  and  give 
tbe  same  product.  Every  similar  example  may  be  solved  in 
like  manner.     Hence, 

KuLE,  Multiply  the  multiplicand  hy  one  of  tJie  factors  of 
the  multiplier t  and  that  product  hy  another  factor,  and  so  on 
until  all  the  factors  in  the  set  have  been  taken  ;  the  last  prodiLct 
will  he  the  true  product. 

49.  Multiply  %  8.37  by  3G.  Ans.  %  301.32. 

50.  Multiply  %  659  by  56.  53.  Multiply  8356  by  81. 

51.  Multiply     737  by  72.  54.  Multiply  6753  by  49. 
62.  Multiply      967  by  96.  55.  Multiply  7045  by  54. 

64.  To  multiply  by  10,  100,  1000,  or  1  with  any 
number  of  ciphers  annexed. 

Rule.  Annex  as  many  ciphers  to  the  multiplicand  as  there 
are  ciphers  in  the  multiplier,  and  the  numher  so  formed  will 
he  the  product. 

64:.    How  is  a  number  multiplied  by  10  ?    By  100  ?  Why  ? 


46  MULTIPLICATION. 

Note.  The  reason  of  the  rule  is  obvious-.  Annexing  a  cipher  re- 
moves each  figure  in  the  multiplicand  one  place  toward  the  left,  and 
thus  its  value  is  increased  ten  fold  (Art.  18). 

56.  Multiply  74  by  10.  Ans.  740. 

57.  Multiply  357  by  1000.  Ans.  357000. 

58.  What  is  8769  X  100? 

59.  What  is  3568  X  10,000? 

60.  What  is  9806  X  100,000  ? 

65.  To  multiply  by  20,  50,  500,  25000,  or  any 
number,  with  ciphers  at  the  riglit :     , 

EuLE.  Multiply  by  the  significant  figures,  and  to  the  prod- 
uct annex  as  many  ciphers  os  there  are  ciphers  at  the  right  of 
the  significant  figures  of  the  inultipUer. 

61.  Multiply  756  by  30.  Ans.  22680. 

OPERATION. 

7  5  6  This  is  upon  the  principle  of  Art.  63. 

3  0         The  factors  of  30  are  3  and  10.     Having 

multiplied  by  3,  the  product  is  multiplied  by 

2  2  6  8  0         10  by  annexing  0  (Art.  64). 
Q2,     Multiply  743  by  3500,  using  factors. 

OPERATION. 

;f  7  4  3 

*7  The  factors  of  3500  are  7,  5, 

and  100,  therefore  multiply  fiirst 

5  2  0  1  by  7,  then  by  5,  then  annex  two 

5  0  0  ciphers. 


Product,    2  6  005  00. 

63.  Multiply  5386  by  42000.  Ans.  226212000. 

64.  Multiply  6539  by  240000.  Ans.  1569360000. 

65.  Multiply  0743  by  630000. 

65.    How  is  a  number  multiplied  by  20?    Why? 


MULTIPLICATION.  47 

06.     To  multiply  when  tliere  are   ciphers  at  the 
right  of  both  multiplicand  and  multiplier  : 

EuLE.     Multiply  the  significant  figures  of  the  multiplicand 
by  those  of  the  multiplier,  and  then  annex  as  many  ciphers  to 
the  product  as  there  are  ciphers  at  theright  of  both  factors. 
Q>Q,     Multiply  8000  by  900.  Ans   7200000. 

The  factors  of  8000  are  8  and  1000, 
OPERATION.  and  those  of  900  arc  9  and  100.     Now, 

8  0  0  0  as  it  is  immaterial  in  what  order  the  fac- 

9  0  0  *  tors  are  taken  (Art.  59,  Note),  first  mul- 
tiply  8  by  9,  then  multiply  this* product 

Ans.  7  2  0  0  0  0  0    by  1000  (Art.  64),  and  this  product  by 
•        lOX 
67.     Multiply  730000  by  2900. 

OPERATION. 

730000 
2900 


657 
14G 


Product,     2  117  0  0  0  0  0  0. 

68.  Multiply  37000  by  29000.  Ans.  1073000000. 

69.  730000  by  47000.  Ans.  34310000000. 

70.  17000  by  79000000. 

71.  4500  by  720000.  Ans.  3240000000. 
67.     To  multiply  by  9,  99,  or  any  number  of  9's. 
Rule.     Annex  as  many  O's  to  the  multiplicand  as  there  are 

9's  in  the  multiplier ^  and  from  the  number  so  formed  subtract 
the  multiplicand  ;  the  remainder  will  be  the  product  sought. 

66.    Rule  when  there  are  ciphers  at  the  right  of  both  factors?    Reason  1 
6T.    Rule  for  multiplying  by  9,  99,  999,  etc. ?    Reason? 


48  MULTIPLICATION. 

72.  Multiply  234:  by  99. 

OPERATION. 

2  3  4  0  0  =  100  times  the  multiplicand. 
2  3  4=      1  time  the  multiplicand. 

23  166  =  99  times  the  multiplicand,  Ans. 

73.  Multiply  5379  by  999.  Ans.  5373621. 

74.  Multiply  638  by  9999.  Ans.  6379362. 

75.  Multiply  739  by  99.     By  999. 

Examples  in  the  Foregoing  Principles. 

1.  A  merchant  bought  156  barrels  of  flour  at  $  9  per  bar- 
rel, and  75  barrels  at  $1 2  per  barrel.  He  also  sold  987  bushels 
of  wheat  at  $  2  per  bushel ;  how  much  more  did  he  pay  for  the 
flour  than  he  received  for  the  wheat  ?  Ans.  $  330. 

2.  Two  men  start  from  the  same  place,  and  travel  in  the 
same  direction,  one  at  the  rate  of  48  miles,  and  the  other  36 
miles  per  day  ;  how  far  apart  are  they  at  the  end  of  17  days  ? 

Ans.  204  miles. 

3.  Had  the  men  named  in  Ex.  2  travelled  in  opposite  direc- 
tions, how  far  apart  would  they  have  been  in  31  days  ? 

Ans.  2604  miles. 

4.  A  farmer  killed  2  oxen  weighing  975  pounds  each,  3 
cows  weighing  462  pounds  each,  and  5  swine  weighing  456 
pounds  each  ;  how  much  more  beef  than  pork  had  he  ? 

Ans.  1056  pounds. 

5.  The  President  of  the  United  States  receives  a  salary  of 
$  25000  a  year ;  what  will  he  save  in  a  year  of  365  days,  if 
his  expenses  are  $  60  a  day  ?  Ans.    $  3 100. 

6.  A  man  having  a  journey  of  287  miles  to  perform  in  5 
days,  travels  62  miles  a  day  for  4  days  ;  how  far  must  he  go 
on  the  fifth  day  ? 


MULTIPLICATION.  49 

7.  Bouglat  a  herd  of  25  cows,  paying  as  many 'dollars  for 
each  cow  as  there  were  cows  in  the  herd.  Paid  $  500  in  money, 
and  gave  my  note  for  the  balance ;  what  was  the  amount  for 
which  the  note  was  given?  Ans.  $  125. 

8.  Bought  13  cows  at  $42  each,  and  21  pair  of  oxen  at 
$  87  a  pair  ;  what  did  I  pay  for  all  ?  Ans.  $  2373. 

9.  Sold  3  farms  ;  for  the  first  I  received  $  2345,  for  the 
second,  $  364' less  than  for  the  first,  and  for  the  third,  twice  as 
much  as  for  the  other  2  ;  how  much  did  I  receive  for  the  3 
farms?  Ans.  $  12978. 

10.  The  factors  of  one  number  are  31  and  43,  and  of 
another  29  and  17  ;  what  is  the  difference  of  the  two  numbers? 

11.  A  teacher  receives  8  1200  a  year,  and  pays  $  364  a 
year  for  board,  8  96  for  clothe?,  $75  for  books,  and  $  356  for 
other  expenses  ;  how  much  will  he  save  in  5  years  ? 

12.  A  manufacturer  receives  $37950  in  one  year  for  the 
products  of  a  certain  factory.  For  materials  he  pays  out 
$  15675,  for  labor  $  10369,  for  repairs  of  machinery  $2006; 
how  much  profit  remains  to  him?  Ans.  $  9900. 

13.  If  the  above  manufacturer,  after  paying  out  of  his 
income  $  5  on  every  hundred  dollars  for  United  States  tax, 
$  3  on  every  hundred  for  other  taxes,  expends  $4875  for  the 
support  of  his  family,  how  much  remains?       Ans.  $  4233. 

14.  A  drover  bought  280  head  of  cattle  for  an  average  cost 
of  $  75  per  head,  10  horses  for  $  210  each;  after  deducting 
the  expenses  of  transporting  them  to  the  market  he  found  he 
had  made  $15  per  head  on  the  cattle,  $50  apiece  on  the 
horses,  what  was  the  amount  of  his  profits? 

15.  A  steamboat  makes  300  trips  in  one  season ;  she  carries 
an  average  of  225  passengers  each  trip,  and  75  tons  of  mer- 
chandize. If  the  average  receipts  are  $2  for  each  passenger 
and  $  1  for  each  ton  of  freight,  how  much  money  does  she 
receive  ?  Ans.  $  167500. 


50  DIVISION. 

16.  A  country  merchant  went  to  the  city  to  purchase  goods, 
carrying  with  him  $3000.  He  bought  20  barrels  of  flour,  at 
$  12  per  barrel,  275  gallons  of  molasses,  at  $  1  per  gallon,  a 
box  of  sugar  for  $  178,  two  pieces  of  broadcloth  at  $  56  a  piece ; 
other  dry  goods  to  the  amount  of  $  525,  and  other  groceries 
to  the  amount  of  $  118,  and  a  variety  of  small  goods  to  the 
amount  of  $375.  After  paying  for  these  how  much  money 
had  he  left? 


DIVISION. 

68.  How  many  peaches,  at  2  cents  each,  can  I  buy  for  6 
cents  ?  Ans.  As  many  as  2  cents  is  contained  times  in  6  cents  ; 
therefore  I  can  buy  3  peaehss^  hecaicse  2  cents  is  contained  3 
times  in  6  cents. 

If  12  apples  are  divided  equally  among  3  boys,  how  many 
apples  will  each  boy  have  ?  Ans.  JSach  boy  will  have  4  apples, 
because  if  12  apples  are  divided  into  3  equal  parts  each 
part  is  4  apples.     These  are  questions  in  Division. 

60.  Division  is  the  process  of  finding  how  many  times 
one  number  is  contained  in  another ;  or.  Division  is  the  pro- 
cess of  separating  one  number  into  as  many  equal  parts  as  there 
are  units  in  another  number. 

The  number  to  be  divided  is  called  the  Dividend  ;  the  num- 
ber by  which  to  divide  is  the  Divisor  ;  the  number  of  times 
the  dividend  contains  the  divisor  is  the  Quotient  ;  and,  if  the 
dividend  does  not  contain  the  divisor  an  exact  number  of  times, 
the  joar^  of  the  dividend  that  is  left  is  the  Kemainder. 

Note.  The  remainder,  heing  a  part  of  the  dividend,  is  always 
of  the  same  kind  as  the  dividend. 

68.  Explain  the  Examples  in  Art.  68.  69.  What  is  Division  ?  Another 
definition  ?  What  is  the  Dividend  ?  Divisor  ?  Quotient  ^  Kemamrler  ?  Of 
what  kind  is  the  remainder  ?    Why  ? 


DIVISION. 
DIVISION  TABLE. 


51 


1  in  I 

Once. 

2  in 

2 

Once. 

3 

in 

3 

Once. 

1  iu  2 

Twice. 

2  in 

4 

Twice. 

3 

in 

6 

Twice.   1 

1  in  3 

3  times. 

2  in 

6 

3  times. 

3 

in 

9 

3  times. 

1  in  4: 

4      " 

2  in 

8 

4      " 

3 

in 

12 

4      •< 

1  in  5 

5      " 

2  in 

10 

5      " 

3 

in 

15 

5      -     i 

1  in  6 

6      " 

2  in 

12 

6      " 

3 

in 

18 

6      " 

1  in  7 

7      - 

2  in 

14 

7      " 

3 

in 

21 

7      *' 

1  in  8 

8      " 

2  in 

16 

8      " 

3 

in 

2t 

8      " 

1  in  9 

9      " 

2  in 

18 

9      " 

3 

in 

27 

9      " 

1 
4  in     4 

Once. 

5  in 

5 

Once, 

6 

in 

6 

Once. 

4  in     8 

Twice. 

5  in 

10 

Twice. 

6 

in 

12 

Twice. 

4  in  12 

3  times. 

5  in 

15 

3  times. 

6 

in 

18 

3  times 

4  in  16 

4      " 

5  in 

20 

4      " 

6 

in 

24 

4      " 

4  in  20 

5      " 

5  in 

25 

5      - 

6 

in 

30 

5      " 

4  in  24 

6      " 

5  in 

30 

6      *' 

6 

in 

36 

6      " 

4  in  28 

7      " 

5  in 

35 

7      '* 

6 

in 

42 

7      " 

4  in  32 

8      " 

5  in 

40 

8      •« 

6 

in 

48 

8      " 

4  in  36 

9      - 

5  in 

45 

9      " 

' 

in 

54 

9      " 

7  in     7 

Once. 

8  in 

8 

Once. 

9 

in 

9 

Once. 

7  in  14 

Twice. 

8  in 

16 

Twice. 

9 

in 

18 

Twice. 

1  in  21 

3  times. 

8  in 

24 

3  times. 

9 

in 

27 

3  times 

7  in  28 

4      " 

8  in 

32 

4      *' 

9 

in 

36 

4      " 

7  in  35 

5      " 

8  in 

40 

5      " 

9 

in 

45 

5      " 

7  in  42 

6      " 

8  in 

48 

6      - 

9 

in 

51 

6      " 

7  in  49 

7      " 

8  in 

56 

7      " 

9 

in 

63 

7      " 

7  in  56 

8      - 

8  in 

64 

8      '' 

9 

in 

72 

8      " 

7  in  63 

9      •' 

8  in 

72 

9      '» 

9 

in 

81 

9      " 

52  DIVISION. 


Mental   Exercises. 


Ex.  1.  How  many  oranges,  at  5  cents  apiece,  can  be  bought 
for  15  cents?  Ans.  As  many  as  5  cents  is  contained  times  in 
15  centSf  namely,  3. 

2.  At  5  cents  an  ounce,  how  many  ounces  of  cloves  can  be 
bought  for  30  cents?  Ans.  6. 

3.  At  $  6  a  cord,  how  many  cords  of  wood  can  I  buy  for 
8  24. 

4.  At  $  8  a  ton,  how  many  tons  of  coal  can  I  buy  for  $  24  ? 
For  $40?     For  $56?     For  $  32  ?  Last  Ans.  4. 

5.  In  how  many  weeks,  at  $  9  a  week,  will  a  man  earn  $  27  ? 
$54?     $36?     $63? 

,6.     At  $  9  a  barrel,  how  many  barrels  of  flour  can  I  buy  for 
$45?     For  $81?     For  $63? 

7.  In  how  many  hours  will  a  horse  travel  36  miles  if  he 
travels  6  miles  per  hour  ?     If  9  miles  ?     If  4  miles  ? 

8.  When  blueberries  cost  10  cents  a  quart,  how  many  quarts 
can  be  bought  for  40  cents?     For  70  cents?     90  cents? 

9.  How  many  sheep,  at  $  1 1  apiece,  can  I  buy  for  $  55  ? 
For  $  44  ?     For  $  66  ?  Last  Ans.  6. 

10.  How  many  pounds  of  coffee,  at  12  cents  a  pound,  can  I 
buy  for  36  cents  ?     For  48  cents  ?     For  72  cents  ? 

11.  Two  men,  72  miles  apart,  approach  each  other  at  the 
rate  of  9  miles  per  hour;  in  how  many  hours  will  they  meet? 

12.  I  divided  15  cents  equally  among  5  boys;  how  many 
cents  did  each  boy  receive?  Ans.  If  l^  cents  are  divided 
into  5  equal  parts,  each  part  is  3  cents,  therefore  each  boy 
received  3  cents. 

13.  A  farmer  sold  5  sheep  for  $  45  ;  what  was  their  average 
price  ? 


DIVISION. 


53 


14.  If  9  men  can  cut  54  cords  of  wood  in  a  week,  how 
many  cords  can  1  man  cut  in  the  same  time  ? 

15.  A  pile  of  48  barrels  of  apples  will  exactly  fill  8  equal 
bins;  how  many  barrels  will  eacJh  bin  hold ? 

16.  I  divided  55  cents  equally  among  11  boys;  how 
many  cents  did  each  receive  ? 

17.  A  dairy  woman  has  84  pounds  of  butter  which  she 
wishes  to  divide  equally  among  her  1 2  customers ;  how  many 
pounds  can  she  furnish  each  ? 

18.  If  a  workman  earns  $48  in  one  month  of  4  weeks, 
how  much  does  he  earn  in  one  week  ? 

1 9.  How  much  will  the  above  workman  earn  in  one  day  ? 

20.  How  many  bushels  of  wheat,  at  $  3  per  bushel,  will  it 
take  to  pay  for  15  bushels  of  rye,  at  $  2  per  bushel? 

21.  If  7  suits  of  clothes  can  be  made  from  64  yards  of 
cloth,  how  many  yards  does  it  take  for  one  suit? 

22.  If  the  cloth  for  one  of  the  above  suits  costs  S  21 ,  how 
much  is  that  per  yard  ? 

70,  The  sign  of  division,  -^,  indicates  that  the  number 
before  it  is  to  be  divided  by  the  number  after  it;  thus, 
8  -^  2  r=  4 ;  that  is,  8  divided  by  2  equals  4  ;  or,  2  in  8,  4 
times. 

Ex.  23.     How  many  are  10  -i 


24.  How  many  are    9 

25.  How  many  are  15 

26.  How  many  arc  16 

27.  How  many  are  49 

28.  How  many  are  72 

29.  How  many  are  84  -i-  7  ? 

30.  How  many  are  8 1 


3? 
5? 

8? 
7? 
9? 


70.    Make  the  sign  of  Division  on  the  blackboard 


54 


DIVISION. 


71,.    Keview  frequently  the  following 
Exercises  in  Division. 


No.  1. 

No.  2. 

No.  3. 

No.  4. 

No.  5. 

8- 

r4 

16- 

r  4 

6- 

^3 

35- 

f-5 

so- 

^-6 

18- 

rQ 

27- 

i-3 

12- 

^-4 

42- 

^6 

le  - 

r  8 

15- 

-3 

45- 

r  9 

24- 

'-Q 

72- 

^•8 

36- 

r9 

49- 

-7 

16- 

r2 

42- 

'-1 

24- 

^4 

14- 

-7 

30- 

-6 

32- 

-8 

40- 

h5 

9- 

^9 

20- 

r2 

8- 

-1 

35- 

-7 

48- 

r  8 

18- 

^3 

28- 

r4 

24- 

-8 

36- 

-6 

54- 

r  9 

14- 

^2 

15- 

r-5 

18- 

-2 

25- 

-5 

18- 

r2 

56- 

f-7 

21- 

r3 

No.  6. 

No.  7. 

No.  8. 

No.  9. 

No.  10. 

30- 

-10 

44- 

-11 

56-^ 

-8 

48- 

-12 

1  -. 

-7 

63- 

-9 

72- 

-9 

36-: 

-12 

84- 

-7 

40-: 

-10 

45- 

-5 

54- 

-9 

70-: 

-7 

12- 

-3 

60-: 

-  12 

36- 

-4 

40- 

-8 

12-^ 

-6 

77- 

-11 

81  -: 

-9 

48- 

-6 

4- 

-4 

20  H 

-4 

6- 

-6 

40-^ 

-4 

63- 

-7 

30- 

-3 

27-: 

-9 

60- 

-4 

55^ 

-  11 

64- 

-8 

21- 

-7 

50-: 

-10 

20- 

-5 

80-^ 

-8 

12- 

-2 

65- 

-5 

60-: 

-5 

8  - 

-8 

50^ 

-5 

No.  11. 

No.  12. 

No.  13. 

No.  14. 

No.  15. 

72- 

-12 

88-^ 

-  11 

60- 

^10 

22- 

-  11 

132- 

^-12 

80- 

-10 

70-^ 

-10 

96- 

r  12 

108- 

-12 

50- 

r^ 

77- 

-7 

99^ 

-9 

90- 

^-9 

100- 

-  10 

66  - 

'-& 

60- 

-6 

96- 

-8 

88- 

:-8 

48- 

-4 

110- 

'-  10 

36- 

r3 

33- 

-3 

24- 

:-i2 

28- 

-7 

132  - 

'- 11 

22- 

-2 

24- 

-2 

99- 

:-ii 

120- 

-12 

110- 

'- 11 

12- 

r  1 

84- 

-12 

90- 

'-  10 

121- 

-11 

20- 

MO 

66- 

r  11 

33- 

-  11 

108- 

^9 

120- 

-10 

144- 

'-  12 

DIVISION. 


55 


72.  Division  is  indicated  not  only  by  the  sign  4-,  given 
in  Art.  70,  but  also  by  the  coloUj  thus,  8:2;  also,  by  writing 
the  divisor  before  the  dividend,  with  a  curved  line  between 
them,  thus,  2)8;  also,  by  writing  the  divisor  under  the  divi- 
dend, with  a  line  between  them,  thus,  | ;  each  of  which 
expressions  means  that  8  is  to  be  divided  by  2. 

73.  The  last  mode  of  indicating  division,  sometimes 
called  the  fractional  sigyi,  gives  the  following  compact 

DIVISION  TABLE. 


i  =  l 

1  =  1 

f  =  l 

t=l 

1  =  1 

1=1 

f  =  2 

1  =  2 

1  =  2 

1  =  2 

1^0=2 

¥=2 

?  =  3 

J  =  3 

1  =  3 

V  =  3 

¥  =  3 

¥  =  3 

\  =  i 

1  =  4 

'#  =  4 

'/=4 

^=4 

¥=* 

\  =  5 

^  =  5 

¥  =  5 

^  =  5 

¥  =  5 

¥  =  5 

1=6 

L2  —  6 

¥  =  6 

2/  =6 

3^0  —  6 

¥=6 

1  =  7 

^=7 

V=7 

^  =  7 

¥=7 

V  =  7 

f  =  8 

.J  =8 

¥  =8 

¥=8 

V=8 

V=8 

f=9 

■^3=9 

V=9 

\«  =  9 

V=9 

¥  =  9 

f=l 

1  =  1 

1  =  1 

i«  =  l 

«  =  1 

if  =  l 

V  =  2 

V=2 

¥  =  2 

*a  =  2 

H  =  2 

^J  =  2 

V=3 

¥  =  3 

¥  =  3 

fj  =  3 

3|=3 

11=3 

y  =  4 

¥  =  * 

¥=4 

^§  =  4 

H  =  4 

f|  =  4 

V=5 

4/  =  5 

V  =  5 

iJ  =  5 

H  =  5 

|»=5 

V  =6 

V  =  6 

V=6 

53-  =  6 

ff  =  6 

if  =  6 

V  =  7 

¥  =  7 

V  =  7 

U  =  7 

ii=7 

41  =  7 

y  =8 

V=8 

V=8 

?J  =  8 

ff=8 

11  =  8 

V  =  9 

y  =  9 

V=9 

fj  =  9 

f^=9 

%'=9 

73.  Second  sign  of  Division,  wliat  is  it  ?  Third  mode  of  indicating  Divi- 
sion, what  is  it?  Fourth  method,  what?  T3,  How  are  the  dividcud  and 
divisor  written  in  the  second  Division  Table  ? 


5Q  DIVISION. 


Ex.  31.     How  many  are  ^S  or  24  -^-  6  ?  Ans.  4. 

32.  How  many  are  -3/,  or  35  -f-  5  ?     3^2.^  or  32  ~  8  ? 

33.  How  many  are  ^,  or  18  H-  2  ?     \%  or  28  -^  7  ? 

34.  How  many  are  ^2.^  or  42  4-  6  ?     4^-,  or  49  -^  7  ? 

35.  How  many  are  %3-,  or  63  -^  9  ?     -7/,  or  72  -f-  8  ? 


Written   Exercises. 

74,  To  perform  Short  Division. 

Ex.  1 .  If  7  days  make  a  week,  how  many  weeks  are  there 
in  364  days? 

Having  written  the 'divisor 

OPERATION.  ^^^  dividend  as  in  the  margin, 

Divisor,  7)364  Dividend,     we  first  say,  7  in  36,  5  times  and 

1  remainder ;  set  the  quotient,  5, 

Quotient,  5  2  under  the  6  of  the  dividend,  and 

then  imagine  the  remainder,  1,  placed  before  the  4,  and  say,  7 
in  14,  2  times;  set  the  2  under  the  4,  and  thus  we  find  the 
quotient  to  be,  52, 

75,  This  process,  called  Short  Division,  usually  employed 
when  the  divisor  is  small,  may  be  performed  by  the  following 

KuLE.  Having  set  the  divisor  at  ths  left  of  the  dividend 
with  a  line  between  them,  divide  the  fewest  figures  in  the 
left  of  the  dividend  that  will  contain  the  divisor,  and  set  the 
quotient  under  the  right  hand  figure  taken  in  tJie  divideyid ; 
if  anything  remains,  prefix  it  mentally  to  the  next  figure  in 
the  dividend,  and  divide  the  number  thus  formed  as  before, 
and  so  -proceed  till  all  the  figures  of  tJie  dividend  have  been 
employed. 


75.  "When  is  Short  Division  usually  employed?  Rule?  How  are  di- 
visor and  dividend  written  ?  Which  figures  of  the  dividend  are  used  first  ? 
How  many?  Quotient,  where  set  ?  Remainder,  to  what  is  il  prefixed?  How? 
What  is  done  with  the  number  so  formed  ?    How  far  is  the  process  carried  ? 


DIVISION.  57 


Ex.  2.  3. 

Divisor,     6 )  S  35 1.54  Dividend.       5  )  875 


Quotient,         $58.59  U2 

5.         6.  7.  8.  9. 

Divide  7218.  8127.         6345.  3528.  2576. 

By  8.         3.  5.  9.  7. 

TG.  When  there  is  no  remainder ^  as  in  the  first  nine  exam- 
ples, the  division  is  complete.  The  dividend  is  then  said  to  be 
divisible  by  the  divisor,  and  the  divisor  is  called  an  exact  divisor. 
When  there  is  a  remainder,  as  in  Ex.  10,  the  division  is  incom- 
plete, and  the  dividend  is  said  to  be  indivisible  by  the  divisor. 

10.  Divide  325  by  7.  Ans.  46f . 

OPERATION. 

Divisor,  7)325  Dividend. 

Quotient,         4  6  . .  3  Remainder. 

Note  1.  The  remainder  is  often  written  over  the  divisor  in  the 
fractional  form  (Art.  73),  and  the  fraction  is  annexed  to  the  quo- 
tient; thus,  the  answer  in  Ex.  10  is  written  4G|^  which,  expressed  in 
words,  is  forty-six  and  three-sevenths.  Other  fractions  are  read  in 
a  similar  manner;  thus,  ^  means  one-half;  J  one-third;  |  twO' 
thirds;  ^Jive-ninths  ;  etc. 

Note  2.  The  remainder,  when  written  in  a  fractional  form  as  a 
part  of  the  quotient,  becomes  like  the  quotient. 

11.  Divide  6276  by  5.     Ans.  1255,  and  1  remainder. 


12. 

Divide  8765  by  5. 

18. 

Divide  7358  by  7. 

13. 

Divide  4823  by  8. 

19. 

Divide  8454  by  9. 

14. 

Divide  6358  by  6. 

20. 

Divide  8684  by  4. 

15. 

Divide  7296  by  2. 

21. 

Divide  $6.84  by  4. 

16. 

Divide  2594  by  3. 

22. 

Divide  $985  by  5. 

17. 

Divide  7828  by  4. 

23. 

Divide  $9.85  by  5. 

76.  "When  is  the  division  complete  ?  When  is  one  number  divisible  by  an- 
other ?  What  is  aa  exact  divisor  ?  When  is  one  number  indivisible  by  another  ? 
How  is  the  remainder  often  written  ?   The  fraction  where  pla  -cd  ? 


58  DIVISION. 

24.  How  many  pounds  of  sugar,  at  9  cents  per  pound,  can 
be  bought  for  $35.64  ?  Ans.  396. 

25.  How  many  barrels  of  flour,  at  $  8  a  barrel,  can  be 
bought  for  $5368? 

26.  If  6  shillings  make   a  dollar,  how  many  dollars  are 
there  in  7416  shillings?  Ans.  1236. 

27.  If  4  weeks  make  a  month,  how  many  months  are  there 
in  3716  weeks? 

28.  How  many  oranges,  at  6  cents  apiece,  can  you  buy  for 
$35.64?  Ans.  594. 

29.  If   7328  marbles  are  divided  equally  among  8  boys, 
how  many  marbles  will  each  boy  receive  ?  Ans.  916. 

30.  If  an  estate,  worth  $16,492  dollars,  is  divided  equally 
among  7  children,  how  many  dollars  does  each  child  receive  ? 

31.  Divide  two  thousand  one  hundred  and  forty-two  by  six. 

32.  A  mile  is  equal  to  5280  feet ;  how  many  steps  of  3  feet 
each  will  a  man  take  in  walking  a  mile  ? 

77.     To  perform  Long  Division  : 

33.  Divide  4654  by  13.  Ans.  358. 
OPERATION.               Having  set  the  divisor  and  dividend 

13')4654C3  58     ^^^°  Short  Division,  draw  a  curve  at  the 
3  9  right  of  the  dividend,  and  then  say,  13 

in  46,  3  times,  and  set  the  3  at  the  right 

7  5  of  the  dividend.     Then  multiply  the  di- 

".*^  visor  by  the  quotient,  3,  and  set  the  pro- 

7TT  duct,  39,  under  the  46  of  the  dividend, 

J  Q  ^  and  subtract  the  39  from  the  46.    To  the 

remainder,  7,  annex  5,  the  next  figure  of 

0  the  dividend,  so  forming  a  new  partial 

dividend,  75,  and  then  say,  13  in  75,  5  times,  and  set  the  5 
as  the  next  figure  of  the  quotient.     Multiply  the  divisor  by 

77.    Explain  £x.  33.    Of  what  order  is  any  quotient  figure  ?    Illustrate. 


DIVISION.  59 

tliis  new  quotient-figure,  and  subtract  the  product  from  the 
partial  dividend.  Proceed  ia  this  manner  until  the  whole 
dividend  has  been  divided;  the  entire  quotient  is  358. 

Every  quotient-jig  are  is  of  the  same  orcler  as  the  right-hand 
figure  of  the  dividend  used  in  obtaining  that  quotient-figure  ; 
thus  in  Ex.  33,  the  46  of  the  dividend  is  hundreds,  and  the  3 
of  the  quotient  is  also  hundreds ;  the  75  is  tens  and  the  5  of 
quotient  is  also  tens  ;  etc. 

78.  This  process,  called  Long  Division,  usually  employed 
when  the  divisor  is  large,  may  be  performed  by  the  following 

EuLE  1.  Write  the  divisor  and  dividend  as  in  Short 
Division. 

2.  Divide  the  smallest  number  of  figures  in  the  left  of  the 
dividend  that  will  contain  the  divisor,  and  set  the  result  as  the 
first  figure  of  tJie  quotient  at  the  right  of  the  dividend. 

3.  Multiply  the  divisor  by  the  quotient  figure,  and  set  the 
product  under  that  part  of  the  dividend  taken. 

4.  Subtract  the  product  from  the  figures  over  it,  and  to 
tlie  remxdnder  annex  the  next  figure  of  the  dividend  for  a 
new  partial  dividend. 

5.  Divide,  and  proceed  as  before,  until  the  whole  dividend 
has  been  divided. 

Note  1.  It  will  be  seen  that  the  process  of  dividing  consists  of 
four  distinct  steps,  viz. :  first,  to  seek'  a  quotient  figure ;  second, 
multiply ;  third,  subtract ;  and,  fourth,  form  a  new  partial  dividend 
by  annexing  the  next  figure  of  the  dividend  to  the  remainder. 

Note  2.  If  any  partial  dividend  will  not  contain  the  divisor,  0 
must  be  placed  in  the  quotient,  and  another  figure  brought  down 
and  annexed  to  the  dividend. 

78.  When  is  Long  Division  employed?  Give  the  rule  for  Long  Division. 
How  many  steps  in  dividing  ?    What  are  they  ?    Repeat  Note  2. 


60  DIVISION. 

Note  3.  If  the  product  of  the  divisor  multiplied  by  the  quotient 
figure  is  greater  than  the  partial  dividend,  the  quotient  figure  is  too 
large,  and  must  be  diminished. 

Note  4.  If  the  remainder  equals  or  exceeds  the  divisor,  the 
quotient  is  too  small,  and  must  be  increased. 

79.  In  the  same  manner  solve  the  following  examples, 
dividing  each  upper  number  by  the  one  under  it  in  each  ex- 
ample ;  also,  in  the  same  manner,  as  suggested  by  the  signs, 

o,    /Divide  18564-^156.                       Ans.  119. 

"^•(By  1092 -^    12. 

o.    (Divide  24453-^143.                       Ans.  171. 

"^^•■[By.  1287 -^    11. 

Oft  j  Divide  20995  -^  221.                         Ans.  95. 

'^^•(By  1105^    13. 

07  j  Divide  143405  -^  989.  Ans.  145. 

"^^•(By  4945 -^    23. 

80.  Division  is  the  reverse  of  multiplication.  In  mul- 
tiplication, the  two  factors  are  given,  and  the  product  is 
required ;  in  division  the  product  and  one  factor  are  given, 
and  the  other  factor  is  required.  The  dividend  is  the  product, 
and  the  divisor  and  quotient  are  the  factors ;  thus, 

IX  MULTIPLICATION.  IN  DIVISION. 

Factors,  Product.  Dividend,       Divisor,      Quotient. 

5    X    4   =    20  20     -^     5     z=     4 

Or,  20     -f-    4     =     5 
Hence  the  following 

Proof.  Multiply  the  divisor  hy  the  quotient,  and  to  the  prod- 
uct add  the  remainder  ;  the  sum  should  be  the  dividend. 

78.  Repeat  Note  3.  Note  4.  80.  What  is  said  of  Division  and  Multipli- 
cation ?  In  Multiplication  what  is  given  ?  What  required  'i  In  Division  what 
i  s  given  ?    Required  ?    How  is  Division  proved  ? 


DIVISION. 


61 


38.     Divide  2537  by  53. 

OPERATION. 

53)2537(4  7 
2  1  2 


PROOF. 

5  3  Divisor. 
4  7  Quotient 


4  1  7 

3  7  1 

3  7  1 

2  1  2 

4  6  Remainder. 

T?^Tn'lTT>'^'i^                A     A 

X.W  w  1 1  lkK  JH-T 

IVAUX,             i    yj 

2  5  3  7   Dividend. 

39. 

40. 

43] 

»87349(2031 

4  7)  9  43  4  54  ( 

;20073 

8  6 

94 

1  34 

3  4  5 

1  29 

32  9 

6  9 

1  6  4 

43 

1  4  1 

Remainder,  1  6 

Eemainder,  2  3 

Quotients. 

Rem, 

41. 

Divide  6384  by  79. 

80. 

64. 

42. 

Divide  7639  by  83. 

92, 

3. 

43. 

Divide  18805 

by  37. 

44. 

Divide  116092  by  29. 

4003, 

5. 

45. 

Divide  47065 

by  231. 

46. 

Divide  29768 

by  123. 

242, 

2. 

47. 

Divide  17693 

by  149. 

48. 

•  Divide  98074 

by  483. 

203, 

25. 

49. 

Divide  69847 

by  348. 

50. 

A  farm  containing  327 

acres,  was  bought  for  $  37605  ; 

what  was  the  price  per  acre  ? 


Ans.  $115. 


62  DIVISION. 

5 1.  Divide  six  thousand  eight  hundred  and  forty- four  acres 
of  land  into  twenty-nine  equal  parts.  Ans.  236  acres. 

52.  A  drover  paid  $2331  for   37   oxen;    what  was  the 
average  price  per  ox  ?  Ans.  $63. 

53.  The  product  of  two  numbers  is  35068765,  and  one  of 
the   numbers   is   8765;    what   is   the   other  number? 

Ans.  4001. 

54.  In  how  many  days  will  a  steamboat  sail  11352  miles, 
if  she  sails  264  miles  per  day  ? 

55.  If  a  railroad  359  miles  long   cost  $  3545484,    what 
was  the  average  cost  per  mile  ?  Ans.  $  9876. 


Contractions. 

81.     To  divide  by  a  composite  number. 

66.     Divide  $  1855  equally  among  35  men. 

OPERATION.  The  35  men  may 

'35  =  7  X  5.  ^®  separated  into  7 

1st  Factor,  7  )  $  1  8  5  5  Dividend.  groups  of  5  men  each. 


Then  dividing  by  7 
2d  Factor,      5  )  $  2  6  5  1st  Quotient,     gives  $265  for  each 

TTT  rx,       ^      .        group,  and  dividing 
$  5  3  True  Quotient.  ^      ^  ^ 

the  $  265  by  5  gives  $  53  for  each  man. 

Note.  When  a  composite  number  is  made  up  of  different  sets  of 
factors,  as  in  Ex.  67,  it  is  immaterial  which  set  is  taken.  It  is  also 
immaterial  in  what  order  the  factors  are  taken. 


81.    Rule  for  dividing  by  a  composite  number  ?    Is  it  material  which  factor 
of  the  divisor  is  used  first,  or  which  set  of  factors  is  employed  ? 


divisio:n.  63 

57.     Divide  10656  by  288. 

288  =  4X6X12  =  6X6X8  =  8X3X12,  etc. 

FIRST  OPERATION.  SECOND  OPERATION. 

4)10656  6)10656 


6)2664  6)1776 


12)444  8)296 

3  7  Ans     =  3  7  Ans. 

From  these  examples  we  have  the  following 

KuLE.  jyivide  the  dividend  hy  one  factor  of  the  divisor,  and 
the  quotient  so  obtained  by  another  factor^  and  so  on  till  all  the 
factors  of  the  set  have  been  used.  The  last  quotient  will  be  the 
true  quotient. 

58.  Divide  20088  by  24  ;  =  6  X  4.  Ans.  837. 

59.  Divide  8445  by  15. 

60.  Divide  23296  by  32.  Ans.  T28. 

61.  Divide  26568  by  12. 

62.  Divide  22720  by  64.  Ans.  355. 

63.  Divide  33696  by  144;   =  12  X  12. 

8S.  In  dividing  by  the  factors  of  the  divisor,  there  may 
be  a  remainder,  after  either  or  each  of  the  divisions. 

Should  the  learner  find  a  difficulty  in  determining  the  true 
remainder,  he  has  but  to  remember  that  it  is  always  of  the 
same  kind  as  the  dividend.      (Art.  69,  Note). 

64.  Divide  86  by  21  ;=  7  X  3. 

OPERATION. 

7)86  In  this  example,  as  86  is  the 

true  dividend,  2  is  the  true  re- 

3)^..2Eem.    ^^^^^^ 

Quotient,     4 

sa.  Rule  for  finding  the  true  remainder  when  the  factors  of  the  divisor 
are  used  separately  ?    The  reason  ? 


64  DIVISION. 

65.     Divide  92  by  28 ;  nz  4  X  7. 

OPERATION.  In  this  example,  as  23  is  only 

4)9  2  one-fourth  of  the  true  dividend, 

so  the  remainder,  2,  is  only  one 

7)23  fourth  of  the   true   remainder ; 

therefore  the  true  remainder  is 

Quotient,  3  ..  2  Eem.  2x4  =  8. 

C6.     Divide  527  by  42  ;  z=  6  X  7. 

OPERATION.  From  the  previous  explanations 

6)527  we  see  that  5  our  first  remainder 

here  is  one  part  of  the  true  re- 

7)87  . .  5  Kern,     mainder,  and  that  3,  the  second 

^      .  ~~"       „  ^  remainder,  multiplied  by  6,  the 

Quotient,         1  2  . .  3  Eem.     ^    ,   ,.  .         .    Ji      .i  . 

nist  divisor,  is  the  other  part ; 

that  is,  5  -|-  3  X  6  =  23  ;  is  the  true  remainder.    Similar  rea- 
soning applies  when  there  are  more  than  two  divisors.     Hence, 

To  obtain  the  true  remainder  when  division  is  per- 
formed by  using  the  factors  of  the  divisor : 

EuLB.  3fultiply  each  remainder ^  except  that  left  hy  tJie  first 
dwision,  hy  the  continued  product  of  the  divisors  preceding  that 
which  gave  the  remainders  severally,  and  the  sum  of  the  prod- 
ucts, together  with  the  remainder  left  hy  the  first  division,  will 
he  the  true  remainder. 

KoTE.  When  there  are  but  two  divisors  and  two  remainders,  it 
only  requires  the  addition  of  the  first  remainder  to  tlie  product 
of  the  first  divisor  and  second  remainder. 

67.     Divide  1834  by  35  ;=  5  X  7.   Ans.  Quo.  52,  Eem.  14. 

08.     Divide  15008  by  315  ;  =5X7X9. 

Ans.  Quo.  47,  Eem.  203. 

69.  Divide  7704  by  105  ;  =3X5X7. 

70.  Divide  45621  by  405  ;  =5X9X9. 

Ans.  Quo.  112,  Eem.  261. 

71.  Divide  55242  by  25. 


DIVISION.  65 

83.  To  divide  by  10,  100,  1000,^etc. 

Rule.  Gat  off  by  a  point,  as  many  figures  from  the  right 
hand  of  tJie  dividend  as  there  are  ciphers  in  the  divisor.  The 
.figures  at  the  left  of  the  point  are  tJie  quotient,  and  those  at  the 
right  are  the  remainder. 

72.  Divide  756  by  10.    Ans.  75.6,  ==  75  Quo.  and  6  Eem. 
Note.    The  reason  of  the  rule  is  obvious.     By  taking  away  the 

right-hand  figure,  each  of  the  other  figures  is  brought  one  place 
nearer  to  units,  and  its  value  is  only  one-tenth  as  great  as  before 
(Art.  18),  and  therefore  the  whole  is  divided  by  10.  For  like  rea- 
sons, cutting  of[  two  figures  divides  by  100;  cutting  oflf  three  figures 
divides  by  1000,  etc. 

73.  Divide  4867  by  100.  Ans.     Quo.  48,  Rem.  67. 

74.  Divide  37692  by  1000. 

75.  Divide  5367842  by  1000000. 

•76.     Divide  20687432004  by  1000000000. 

84.  To  divide  by  20,  50,  700,  or  aoy  like  number  : 
Rule.     Gut  off  as  many  figures  from  the  right  of  the  divi- 
dend as  there  are  ciphers  at  the  right  of  the  significant  figures 
of  the  divisor,  and  then  divide  the  remaining  figures  of  the 
dividend  by  the  significant  figures  of  the  divisor. 

Note  1.  This  is  on  the  principle  of  dividing  by  the  factors  of 
the  divisor ;  therefore  the  true  remainder  will  be  found  by  the  rule 
in  Art.  82. 

77.     Divide  74689  by  8000.  Ans.  9  Quo.  and  2689  Rem. 

OPERATiox.  We  divide  by  1000  by  cut- 

8)  7  4.6  8  9  ting  off  689,  which  gives  74 

for  a  quotient  and  689  for  a 

Quotient,  9  ...  2  Rem.  remainder ;  then  divide  74  by 

8,  and  obtain  the  quotient,  9,  and  remainder,  2.  This  remain- 
der, 2,  is  2000,  which,  increased  by  689,  gives  2689  for  the 
true  remainder  (Art.  82). 

83.  Rule  for  dividing  by  10  ?  By  100  ?  Reason  of  rule  ?  84.  Rule  for  divid- 
ing by  20?    By  500?    Reason?    How  is  the  true  remainder  found  ? 


66  DIVISION. 

Note  2.  It  will  be  observed  that  the  true  remainder,  in  all  ex- 
amples like  the  77th,  is  obtained  by  annexing  the  1st  to  the  2d  re- 
mainder. 

78.  Divide  3764  by  200.       Ans.  Quo.,  18,  Rem.,  164. 

79.  Divide  4547  by  400. 

80.  Divide  3876423  by  YOOO.   Ans.  Quo.  553,  Kern.  5423. 

81.  Divide  7943862  by  210000. 

General  Principles  of  Division. 

85.  The  value  of  a  quotient  depends  upon  the  rela- 
tive values  of  the  divisor  and  dividend,  and  not  upon 
their  absolute  values.  This  will  be  seen  by  the  follovv^- 
ing  propositions. 

(1st).  If  the  divisor  remains  unaltered^  multiplying  the 
dividend  hy  any  number  is,  in  effect,  multiplying  the  quotient 
by  the  same  number  ;  thus, 

15  -^  3  irr     5 
4  4 

60-1-3  =  20 
that  is,  multiplying  the  dividend  by  4  multiplies  the  quotient 
by  4. 

(2d),       Dividing  the  dividend  by  any  number  is  dividing 
the  quotient  by  the  same  number  ;  thus, 
24-^-2=12 
3)24 

^-^2=    4=nl2-^-3; 
that  is,  dividing  the  dividend  by  3  divides  the  quotient  by  3. 

85.  Does  the  size  of  the  quotient  depend  upon  the  absolute  size  of  the 
divisor  and  dividend  ?  Upon  what  does  it  depend  ?  VThat  is  the  first  propo- 
sition?   Second?    Third?    Fourth? 


} 


DIVISION.  67 

(Sd).      Multiplying  the  divisor  hy  any  numher  divides  the 
quotient  hy  the  same  numher  ;  thus, 
3  0  -^  2  =  15 
3 

30-^6==    5  =  15-^3; 
that  is,  multiplying  the  divisor  by  3  divides  the  quotient  by  3. 
(4  th).     Dividing  the  divisor  hy  any  numher  multiplies  the 
quotient  hy  the  same  numh3r  ;  thus, 
40  -I-  10=r4 
5)  10 

40  -^     2  =  20  =  4  X  5; 
that  is,  dividing  the  divisor  by  5  multiplies  the  quotient  by  5. 

(5th).  It  follows,  from  (1st)  and  (2d),  that  the  greater 
the  dividend  the  greater  is  the  quotient;  and  the  less  the 
dividend  the  less  the  quotient. 

(6th).  Also,  from,  (3d)  and  (4th),  that  the  greater  tJie 
divisor,  the  less  is  the  quotient ;  and  the  less  the  divisor  the 
greater  the  quotient. 

86.  From  the  illustrations  in  Art.  85  we  see  that 
any  change  in  the  dividend  causes  a  similar  change  In 
the  quotient,  and  that  any  change  in  the  divisor  causes 
an  OPPOSITE  change  in  the  quotient.     Hence, 

(1st),     Multiplying  hoth  dividend  and  divisor  hy  the  same 
numher  does  not  affect  the  quotient ;  thus, 
12-^-3  = 
2       2 

2  4-^6  =  4,  Quotient  unchanged. 

83.  What  follows  from  (Ist)  and  (2d)  ?  What  follows  from  (3d),  (4th)? 
From  (5th),  (6th)  ?  86.  Any  change  in  the  dividend,  how  affects  the  quotient  ? 
Any  change  in  the  divisor,  how?   First  inference  ?   Second?   Third?  Illustrate. 


68  DIVISION. 

(2d).  Dividing  both  dividend  and  divisor  hy  the  same 
number  does  not  affect  the  quotient;  thus, 

20        -^        10:^2 
5)20  6)10 

4       ~         2  z=  2,  Quotient  unchanged. 

(3d).  It  follows,  from  (1st)  and  (2dJ,  that  the  operations 
of  multiplying  and  dividing  hy  the  same  number  cancel^  that  is 
destroy,  each  other ;  for  example, 

If  a  number  be  multiplied  by  any  number,  and  the  product 
be  divided  by  the  multiplier,  the  (juotient  will  be  the  multipli- 
cand; thus, 

8  X  7  =  56  and  56  -^  7  ==  8,  the  multiplicand. 

Also,  if  a  number  be  divided  by  any  number,  and  the  quo- 
tient be  multiplied  by  the  divisor,  the  product  will  be  the 
dividend;  thus, 

15  -h  3  —  5,  and  5  X  3  =  15,  the  dividend. 

87,  These  general  principles  may  be  naore  briefly 
stated  as  follows ; 

(1st).  Multiplying  the  dividend  multiplies  the  quotient; 
and  dividing  the  dividend  divides  the  quotient  (Art.  85,  1*^ 
and  2nd). 

(2d) .  Multiplying  the  divisor  divides  the  quotient ;  and 
dividing  the  divisor  multiplies  the  quotient  (Art.  85,  3c? 
and  ^th) . 

(3d).  Multiplying  both  dividend  and  divisor  by  the  same 
number  ;  or  dividing  both  by  the  same  number,  does  not  affect 
the  quotient  (Art.  86,  1st  and  2d). 

8T,    A  more  brief  statement  of  these  principles :   First  ?   Second  ?    Third  ? 


DIVISION,  69 


CANCELLATION. 

88.  How  many  are  6  times  7  divided  by  6  ? 
OPERATION.  Indicating  the  multiplication  and 

0X7  division  (Art.  73),  we  may  cancel  or 

=  7,  Ans.       strike  out  the  equal  factors,  6  and  6, 

0  from  the  divisor  and  dividend,  and  so 

shorten  the  process  without  changing  the  result. 
How  many  are  7  times  1 2  divided  by  6  ? 
OPERATION.  Separating  the  12  into 

7  X  12         7X2X0  the  two  factors,  2  and  6, 

-^-^^  or, :=  14  Ans.     cancel  the  G  from  divisor 

6  0  and  dividend,  and  there 

is  left  7  times  2,  equal 
to  14,  for  the  quotient. 
'2  This  process  is,  in  ef- 

7  X  t  ^  feet,  the  same  as  the  other. 

=  14,  Ans.         Instead  of  resolving  12 

0  into  the  factors,  2  and  G, 

we  merely  divide  12  by  6,  setting  the  quotient,  2,  over  the  12 ; 
then,  cancelling  the  6  and  12,  the  result  is  7  times  2,  equal  to 
14,  as  before. 

How  many  are  8  times  15  divided  by  12  ? 

First,  reject  or  cancel  the  factor  4 
OPERATION.  fj.^^  I^J^^^  8  and  12,  giving  2,  which 

2         5  is  placed  over  8  and  3,  placed  under 

$  X  3^5  12 ;  thien  cancel  3  in  3,  and  in  15, 

— —  =10,  Ans.  giving  5  to  place  over  1 5 ,  and  we  have 

rt  2  times  5,  equal  to  10,  for  the  result. 

These  examples  arc  solved,  in  part,  by  cancelling.     Hence, 

89.  Cancellatiox  is  a  process  for  shortening  the  solution 

89.    What  is  Cancellation?    On  what  principle  does  it  depend? 


70  DIVISION. 

of  an  example,  by  rejecting,  or  cancelling  the  same  factors 
from  both  divisor  and  dividend. 

It  depends  on  the  principle  (Art.  86,  2c?),  that  dividing 
both  dividend  and  divisor  hy  the  same  number  does  not  affect 
the  quotieiit. 

Ex.  1.     Divide  8  X  3  X  10  X  63  by  4  X  5  X  7. 

OPERATION.  '  We  cancel  4  in  8,  giv- 

2  2         9  iiig  2  ;  5  in  10  giving  2, 

§  X  3  X  3:0  X  03  and  7  in  63,  giving  9. 

.^  .  ^ry ==  10^'  ^°3-     Then,  2X3X2X9 

^^^^f^  ^  108,  the  Ans. 

2.  Divide  6  X  21  X  15  X  11  by  18  X  7  X  5.    Ans.  33. 

3.  Divide  9  X  14  X  26  X  8  by  3  X  21  X  13  X  4. 

4.  How  many  cords  of  wood,  at  $6  a  cord,  will  pay  for  5 
tons  of  hay,  at  §12  a  ton? 

6.  How  many  tons  of  hay,  at  $15  a  ton,  will  pay  for  4 
acres  of  land,  at  $  45  an  acre?" 

6.  How  many  pounds  of  butter,  at  33  cents  a  pound,  must 
be  given  for  3  boxes  of  raisins,  each  containing  22  pounds,  at 
15  cents  a  pound?  Ans.  30. 

7.  How  many  pieces  of  cloth,  containing  32  yards  each, 
at  $3  per  yard,  will  pay  for  48  barrels  of  flour,  at  $12  per 
barrel?  •  Ans.  6. 

8.  How  much  sugar,  at  15  cents  a  pound,  will  be  required 
to  pay  for  3  boxes  of  lemons,  containing  305  lemons  each,  at 
4  cents  apiece.  Ans.  244. 

9.  How  many  boxes  of  tea,  each  containing  45  pounds, 
at  66  cents  a  pound,  must  be  given  for  15  boxes  of  sugar, 
each  containing  220  pounds,  at  18  cents  a  pound? 

10.  How  many  bags  of  corn,  each  containing  2  bushels,  at 
96  cents  a  bushel,  will  pay  for  128  bags  of  oats,  each  con- 
taining 3  bushels,  at  47  cents  a  bushel?  Ans.  94. 


DIVISION.  71 


Eeyiew  and  Test  Questions. 

90.     Let  the  pupil  answer  the  following  questions, 
illustrating  them  with  his  own  original  examples  : 

1.  How  will  you  find  the  sum  of  several  given  numbers? 

2.  How  will   you  find  the  difierence  between  two  given 
numbers  ? 

3.  How  will  you  find  the  subtrahend  when  the  minuend 
and  remainder  are  given  ? 

4.  How  will  you  find  the  minuend  when  the  subtrahend 
and  remainder  are  given  ? 

6.  How  will  you  find  the  remainder,  when  the  minuend 
and  subtrahend  are  given  ? 

6.  When  the  sum  of  several  numbers  and  all  of  the  numbers 
except  one  are  given,  how  do  you  find  that  one  ? 

7.  When  the  multiplicand  and  multiplier  are  given,  how 
can  you  fiiid  the  product  ? 

8.  When  the  multiplicand  and  product  are  given,  how  can 
you  fiud  the  multiplier  ? 

9.  When  the  multiplier  and  product  are  given,  how  can 
you  find  the  multiplicand  ? 

10.  How  do  you  find  the  quotient,  when  the  dividend  and 
divisor  are  given  ? 

11.  How  do  you  find  the  divisor,  when  the  dividend  and  quo- 
tient are  given  ? 

12.  How  do  you  find  the  dividend,  when  the  divisor  and 
quotient  are  given  ? 

13.  How  do  you  find  the  dividend,  when  the  quotient,  divi- 
sor, and  remainder  are  given  ? 

14.  How  do  you  find    the  divisor,  when  the  dividend, 
quotient  and  remainder  are  given  ? 


72  DIVISION* 


Examples  in  the  Foregoing  Principles. 

1.  A  boy  sold  a  sled  for  $2.00  and  in  payment  received 
50  cents  in  money,  5  pineapples  at  20  cents  each,  and  the  re- 
mainder in  oranges  at  5  cents  each ;  how  many  oranges  did 
he  receive?  Ans.  10. 

2.  If  2  barrels  of  flour  will  pay  for  5  yards  of  broadcloth, 
how  many  barrels  of  flour  will  pay  for  3  times  5  yards  of  broad- 
cloth? 

3.  How  many  barrels  of  apples,  at  $  3  a  barrel,  must  be 
given  for  6  yards  of  flannel,  at  $  2  a  yard  ? 

4.  A  speculator  bought  80  acres  of  land  at  $  Vo  per  acre, 
and  sold  the  whole  for  S  6720  ;  how  much  did  he  gain  by  the 
transactions  ?     How  much  per  acre  ?         First  Ans.  $  720. 

5.  Bought  160  acres  of  land  for  $4000,  and  sold  it  at 
$20  per  acre;  did  I  gain  or  lose?  How  much?  How  much 
per  acre  ? 

6.  If  2  men  build  16  rods  of  wall  in  4  days,  in  how  many 
days  will  5  men  build  50  rods  ?  Ans.  5. 

t.  How  many  miles  per  hour  must  a  steamboat  sail  to 
cross  the  Atlantic,  2880  miles,  in  10  days  of  24  hours 
each?  Ans.  12. 

8.  The  product  of  two  factors  is  595  ;  one  of  the  factors  is 
17  ;  what  is  the  other  ? 

9.  The  product  of  three  factors  is  9177;  two  of  the  factors 
arc  21  and  19  ;  what  is  the  other?  Ans.  23. 

10.  The  divisor  is  18,  and  the  quotient  13  ;  what  is  the  divi- 
dend? 

'  11.  The  divisor  is  23,  the  quotient  37,  and  the  remainder  19  ; 
what  is  the  dividend  ?  Ans.  870. 


DIVISION.  73 

12.  The  first  of  three  numbers  is  8,  the  second  is  4  times  the 
first,  and  the  third  is  3  times  the  sum  of  the  other  two  ;  what 
is  the  difi"erence  between  the  first  and  third  ?         Ans.  112. 

13.  In  how  many  days  of  24  hours  each,  will  a  ship  cross 
the  Atlantic',  2880  miles,  if  she  sails  12  miles  per  hour  ? 

14.  If  I  receive  $  80  and  spend  S  55  por  month,  in  how 
many  years  of  12  months  each  shall  I  save  $  1800  ? 

15.  Bought  87  yards  of  cloth,  at  $  4  per  yard,  and  paid  S  200 
in  money  and  the  rest  in  wheat  at  S  2  per  bushel ;  how  many 
bushels  of  wheat  did  it  take  ?  Ans.  74. 

16.  The  subtrahend  is  3762,  and  the  remainder  is  2657  ; 
what  is  the  minuend  ? 

17.  The  minuend  is  8063,  and  the  remainder  is  5604  ;  what 
is  the  subtrahend?  Ans.  2459. 

18.  The  greater  of  two  numbers  is  8327,  and  the  diflfcrence 
is  5364  ;  what  is  the  less  number  ?  Ans.  2963. 

19.  The  sum  of  two  numbers  is  5836,  and  the  less  number 
is  2467  ;  what  is  the  difference  between  the  two  numbers  ? 

20.  A  man  having  engaged  to  work  12  months  for  S  432, 
left  his  employer  at  the  end  of  9  months  ;  at  the  rate  agreed 
upon,  what  should  he  receive  ?  Ans.  $  324. 

21  A  merchant  received  $376  on  Monday,  $567  on  Tues- 
day, $487  on  Wednesday,  $684  on  Thursday,  $293  on  Friday, 
and  $  857  on  Saturday ;  what  were  his  average  receipts  per  day? 

22.  If  732  is  multiplied  by  27  and  the  product  divided  by 
36,  what  is  the  quotient  ?  Ans.  549. 

23.  Bought  175  barrels  of  flour  for  $  1750,  and  sold  86  bar- 
rels of  it  at  $  12  a  barrel,  and  the  remainder  at  $  9  a  barrel  ; 
did  I  gain  or  lose  ?     How  much  ? 

24.  How  many  are  376  +  874  +  563  —  937? 

25.  How  many  are  384  +  562  -j-  1728  -^  191  ? 


74  REDUCTION . 


DENOMINATE  NUMBEKS  AND  EEDUCTION. 

91,     All  numbers  are  either  concrete  or  abstract. 

A  Concrete  Number  is  one  that  is  applied  to  a  particular 
object ;  as  6  books,  4  men,  7  days,  3  rods.  A  concrete  num- 
ber is  often  called  a  Denominate  Number,  because  it  denomi- 
nates or  names  the  thing  to  which  it  is  applied. 

An  Abstract  Number  is  one  that  is  not  applied  to  any  par- 
ticular object;  as  6,  9,  23. 

93,     All  numbers  are  either  simple  or  compound. 
A  Simple  Number  consists  of  but  one  kind,  and  may  be 
abstract  or  concrete  j  as  2,  $4,  10  miles,  3  pounds. 

A  Compound  Number  consists  of  two  or  more  denominations, 
and  is  necessarily  concrete  ;  as  4  days  and  7  hours  ;  3  pecks, 
5  quarts,  and  1  pint ;  8  rods,  4  yards,  2  feet,  and  10  inches. 

Note  1.  The  several  parts  of  a  compound  number,  though  of  dif' 
ferent  denominations,  are  yet  of  the  same  general  nature  ;  thus,  2 
weeks,  3  days,  and  6  hours  are  similar  quantities,  and  constitute  a 
compound  number;  but  2  weeks,  3  miles,  and  6  quarts  are  unlike 
IN  THEIR  NATURE,  and  do  NOT  Constitute  a  compound  number. 

Note  2.  The  first  division  of  each  of  the  following  tables  should 
be  thorovghly  committed  to  memory.  The  second  division  is  designed 
for  reference. 


91.  What  is  a  Concrete  Number  ?  What  is  it  often  called  ?  Why?  What 
is  an  Abstract  Number  ? 

95i.  What  is  a  Simple  Number  ?  May  it  be  abstract  ?  Concrete  ?  Illustrate. 
What  is  a  Compound  Number  ?  May  it  be  abstract  ?  Illustrate.  What  is  said 
of  the  different  denominations  of  a  compound  number  ?  Is  Viis  a  compound 
number  :  3  rods,  2  pecks,  and  6  pounds  ?  Why  ?  What  is  said  of  the  first 
divisioa  of  the  following  tables  ?    What  of  the  second  ? 


1  Shilling,    " 
1  Pound,      " 

s. 
£ 

qr. 
=       4 

=     48 

—   960 

EEDUCTION.  75 

ENGLISH  MONEY. 
03«     English  Money  is  the  Currency  of  Great  Britain. 

TABLE. 

4  Farthings  (far.  or  qr.)  make  1  Penny,  marked  d. 

12  Pence  '* 

20  Shillings  •* 

d. 

s.  1 

£  1      =        12 

1       z=   20      =     240 

Note.  The  numbers  employed  as  multipliers  and  divisors  in 
reducing  a  Compound  Number  are  called  a  Scale  ;  thus,  in  reducing 
English  Money,  the  Descending  Scale  is  20,  12,  and  4 ;  and  the 
Ascending  Scale  is  4,  12,  and  20. 

Ex.  1.     In  7£  Is.  6d.  8qr.  how  many  farthings  ? 


Multiply  the  7  by  20  to  change 
the  pounds  to  shillings ;  to  the 
product,  140,  add  the  Is.  givenjn 
the  example,  and  the  result  is 
141s.  ;  then  multiply  the  141  by 
12  to  change  the  shillings  to  pence  ; 
to  the  product,  1692,  add  the  6d. 
in  the  example,  and  the  result  is 
1 6  9  8d. ;  so  proceed  till  the  example 
is  solved. 


OPERATION. 

£         8. 

d. 

qr. 

7     1 

6 

3 

20 

14  1s. 

1  2 

1  6  9  8  d. 

4 

6  7  9  5  qr.,  Ans. 


93.  What  is  English  Money  ?  Repeat  the  Table.  What  are  the  multi- 
pliers and  divisors  used  in  reducing  a  compound  number  called  ?  What  is 
the  descending  scale  in  English  Money  ?  What  the  ascending  scale  ?  Explain 
Example  1,    Explain  Ex.  2. 


76  REDUCTION. 

Ex.  2.     Change  6795  qr.  to  pence,  shillings,  and  pounds. 

OPERATION.  First  divide  by  4  to  change  the 

4  )  6  7  9  5  qr.  farthings  to  pence,  giving  1698d. 

and  3qrs.  ;  then  divide  the  1698 

12)1698  d.  -|-  Sqr.     by   12  to  change  the  pence  to 

shillings,  giving  141s.  and  6d. ; 

2  0  )  1  4  1  s.  +  6d.      then  divide  the    141    by   20  to 

change  the   shillings  to  pounds, 

7  £  4"  Is*      giving  7£    Is.,   and  thus  obtain 
7£  Is.  6d.  3qr.,  Ans. 
These  examples  are  questions  in  Reduction.     Hence, 

04:.  EEDUCiiiON  consists  in  changing  a  number  of  one  de- 
nomination to  a  number  of  another  denomination,  without 
changing  its  value. 

The  process  in  Ex.  1  is  called  Reduction  Descending^  because 
higher  denominations  are  changed  to  lower.     Hence, 

95,  Eeduction  Descending  consists  in  changing  a  num- 
ber from  a  higher  to  a  lower  denomination,  and  may  be  per- 
fopmed  by  the  following 

EuLE.  Multiply  the  highest  denomination  given,  by  the  num- 
ber it  takes  of  the  next  lower  denomination  to  make  one  of  this 
higher f  and  to  the  product  add  the  number  of  the  lower  denom- 
ination ;  multiply  this  sum  by  the  number  it  takes  of  the  next 
lower  denomination  to  make  one  of  this  ;  add  as  before,  and 
so  proceed  till  the  number  is  brought  to  the  denomination  re- 
q  'lired. 

06,  The  process  in  Ex.  2  is  called  Reduction  Ascending, 
because  lower  denominations  are  changed  to  higher.     Hence, 

94.  What  is  Reduction  ?  What  is  the  process  in  Ex.  1  called  ?  Why  ? 
95.  What  is  lleduction  Descending  ?  Rule  for  performing  it  ?  96,  What 
is  the  process  in  Ex.  2  called  ?    Why  ? 


REDUCTION.  77 

Eeduction  Ascending  consists  in  changing  a  number  from 
a  lower  to  a  higher  denomination,  and  may  be  performed  by 
tte  following 

EuLB.  Divide  the  given  number  by  the  number  it  takes  of 
that  denomination  to  make  one  of  the  next  higher  ;  divide  the 
quotient  by  the  number  it  takes  of  that  denomination  to  niake 
one  of  the  next  higher ,  and  so  proceed  till  the  number  is 
brought  to  the  denomination  required.  The  last  quotient,  to- 
gether with  the  several  remainders  (Art.  69,  Kote),  will  be  the 
answer. 

97.  The  processes  in  Eeduction  Ascending  and  Eeduction 
Descending  prove  each  other,  as  will  be  seen  in  Examples  3  and 
4.  In  the  same  manner  let  the  pupil  prove  all  the  examples 
in  Eeduction,  and  the  answers,  for  this  purpose,  will  be  omit- 
ted in  the  book. 

Ex.  3.  How  many  farthings  Ex.  4.  Eeduce  15542  qr.  to 
in  IQ£  3s.  9d.  2qr. ?  pence,  shillings,  etc. 


OPERATION 

16£     3s. 
20 

323  s. 
12 

9d. 

2qr. 

OPERATION. 

4  )  1  5  5  4  2  qr. 
1  2  )  3  8  8  5  d.  +  2qr. 
2  0  )  3  2  3  s.  +  9d. 

3  8  8  5  d. 
4 

1  6  £  +  3s. 
Ans.  16£  3s.  9d.  2qr. 

15  5  4  2  qr.,  Ans. 

Note  1.  In  solving  Ex.  3,  the  several  numbers  of  the  lower  de- 
nominations are  added  mentally j  and  only  the  results  are  written  ; 
.tlius,  20  times  16  are  320,  and  the  3s.  added  give  323s.     Then  mul- 

96.  What  is  Reduction  Ascending  f  Rule  for  performing  it  ?  97.  How 
are  processes  in  Reduction  proved  ?  In  solving  Ex.  3,  how  are  the  numbers  of 
the  lower  denominations  added  ? 


78  REDUCTION. 

tiplying  the  323  by  12,  and  adding  the  9d.,  we  have  3886d.  Finally, 
multiplying  the  3885  by  4,  and  adding  the  2qr.,  we  have  15542qr. 
which  is  the  Ans. 

Note  2.  In  solving  Ex.  4,  and  other  examples  in  Reduction  As- 
cending, if  any  divisor  is  so  large  that  the  work  is  not  easily  done 
by  Short  Division,  the  numbers  may  be  taken  upon  the  slate  and  the 
work  done  by  Long  Division. 

5.  Reduce  27£  16s.  lid.  Iqr.  to  farthings. 

6.  Ecduce  17375  qrs.  to  pence,  shillings,  and  pounds. 

7.  Eeduce  54£  9s.  3qr.  to  farthing.^. 

8.  Ecduce  25£  3d,  to  farthings. 

9.  Eeduce  12497qr.  to  higher  denominations. 

10.  Ecduce  23445  qr.  to  higher  denominations. 

11.  A  bookseller  received  from  London  fifty  Oxford  Bibles. 
The  lot  cost  him  6£  5s.,  how  much  was  that  apiece? 

TROY  WEIGHT. 
08.     Troy  Weight  is  used  in  weighing  gold,  silver,  and 
precious  stones. 

TABLE. 

24  Grains  (gr.)         make 
20  Pennyweights  ** 

12  Ounces  " 


1  Pennywei 

ght,     dwt. 

1  Ounce, 

oz. 

1  Pound, 

lb. 

dwt 

gr- 

1         = 

24 

20        = 

480 

240          =: 

6760 

oz. 

lb.  1  =: 

1         =         12        = 

Scale. — Descending,  12,  20,  24;  ascending,  24,  20,  12. 

97.  In  Ex.  4,  how  is  the  work  done?  98.  For  what  is  Troy  "Weight 
used  ?  Repeat  the  table.  What  is  the  descending  scale  ?  Ascending  ?  Ex- 
plain Ex.  1. 


REDUCTIOX 


79 


Ex.  1.  How  many  grains  Ex.  2.  Reduce  4  5  9  5  4  gr. 

in  71b.  lloz.  14dwt.  18gr.  ?  to  pounds,  ounces,  etc. 

OPERATIOX.  OPERATION. 

7  lb.  lloz.  Udwt.  18gr.  2  4  )  4  5  9  5  4  gr. 

1  2  

2  0)19  14dwt. +  18gr. 


12)  9  5oz.  +  14dwt. 
71b.  +  lloz. 
Ans.  71b.  lloz.  14dwt.  18gr. 


4  5  9  5  4  gr.,  Ans, 

Note  1.  In  reducmg  the  pennyweights  to  grains  in  Ex.  1,  we  first 
multiply  the  1914  by  4  and  add  the  18gr.,  giving  7674 ;  then  multiply 
the  1914  by  the  2  tens,  giving  3828  tens ;  and  finally  add  the  results, 
giving  45954gr.,  Ans. 

3.  Reduce  61b.  4oz.  ISdwt.  28gr.  to  grains. 

4.  Reduce  181b.  lloz.  6dwt.  19gr.  to  grains. 

5.  Reduce  53649gr.  to  pennyweights,  ounces,  etc. 

6.  Reduce  63594gr.  to  higher  denominations. 

7.  Reduce  151b.  6dwt.  to  grains. 

8.  How  many  spoons,  each  weighing  2oz.  3dwt.  18gr.,  can 


9  5  oz. 
20 

1  9  1  4  dwt. 
24 

7  6  74 

3  82  8 

be  made  from  lib.  loz.  2dwt.  12gr.  of  silver? 


Ans.  6. 


9.  A  jeweller  made  6oz.  7dwt.  12gr.  of  gold  into  rings, 
which  weighed  3dwt.  13gr.  each;  how  many  rings  did  he 
make? 

Note  2.  In  performing  Exs.  8  and  9,  and  similar  examples,  both 
of  the  given  quantities  must  first  be  reduced  to  the  lowest  denomin- 
ation mentioned  in  either. 


80  REDUCTION. 

APOTHECAKIES'   WEIGHT. 
99.     Apothecaries'  Weight  is  used  in  mixing  or  com- 
pounding medicines ;  but  medicines  are  bought  and  sold  by 
Avoirdupois  Weight, 

TABLE. 


20  Grains  (gr.) 

make 

1  Scruple 

,  so. 

or  B 

3  Scruples 

<{ 

1  Dram, 

dr. 

or  5 

8  Drams 

tt 

1  Ounce, 

oz. 

or  § 

12  Ounces 

tt 

1  Pound, 
so. 

lb. 

or  lb 
gr. 

dr. 

1 

z=z 

20 

oz. 

1 

=          3 

=■ 

60 

lb.                  1       — 

8 

—         24 

= 

480 

I       —       \2       = 

96 

=       288 

— 

5760 

Scale.     Descending,  12,  8,  3,  20;  Ascending,  20,  3,  8, 12. 

Note.     The  pound,  ouDce,  and  grain,  in  Apothecaries'  and  Troy 
Weight  are  equal,  but  the  ounce  is  differently  subdivided. 

1.  Beduce  21b3i53  19  18gr.  to  grains. 

2.  Eeduce  13298gr.  to  scruples,  drams,  etc. 

3.  In  51b.  6oz.  5 dr.  2sc.  14gr.  bow  many  grains  ? 

4.  In  3  lb  5  §   33  9  24gr.  how  many  grains  ? 

5.  In  2543 7gr.  bow  many  sciuples,  drams,  etc.  ? 

6.  Eeduce  3764gr.  to  higher  denominations. 

7.  What  quantity  of  medicine  will  an  apothecary  use  in 
preparing  365  prescriptions  of  12  grains  each? 

Ans.  9oz.  Idr. 


99.  For  what  is  Apothecaries'  Weight  used  ?  Eepeat  the  table.  Descend- 
ing scale  ?  Ascending  ?  What  denominations  of  Apothecaries'  Weight  are 
like  those  of  Troy  Weight  ?    What  of  tlie  ounce  ? 


REDUCTION.  81 

AVOIRDUPOIS  WEIGHT. 
100.     Avoirdupois  Weight  is   used   in    weighing  the 
coarser  articles  of  merchandise,  such  as  hay,  cotton,  tea,  sugar, 
copi)er,  iron,  etc. 


TABLE. 

16  Drams  (dr.) 

make 

1  Ounce, 

oz. 

16  Ounces 

ti 

1  Pound, 

lb. 

25  Pounds 

<( 

1   Quarter, 

qr. 

4  Quarters 

ti 

1   Hundredweight, 

cwt. 

20  Hundredweight" 

1  Ton, 

t. 

oz. 

dr. 

lb.                     1     = 

16 

qr. 

1     =          16     = 

256 

cwt.             1 

25     =z         400     = 

6400 

t               1=4 

zm 

100     =       1600     = 

25600 

1     =      20     =     80 

= 

2000     =     32000     =: 

512000 

Scale.  Descending,  20,  4,  25,  16,  16  ;  Ascending,  16, 16, 
25,  4,  20. 

Note  1.  It  was  the  custom  formerly  to  consider  281b.  a  quarter, 
1121b.  a  hundred  weight,  and  22401b.  a  ton ;  but  now  the  usual  prac- 
tice is  in  accordance  with  the  table. 

These  different  tons  are  distinguished  as  the  long  or  gross  ton  = 
22401b.  and  the  short  or  net  ton  =  20001b. 

The  gross  ton  is  still  used  in  the  wholesale  coal  trade ;  also  in  esti- 
mating goods  at  the  U.  S.  custom-houses,  etc. 

Note  2.  A  pound  in  Avoirdupois  Weight  is  equal  to  7000  grains 
in  Apothecaries,  and  Troy  Weight. 


100.  For  what  is  Avoirdupois  Weight  used?  Table?  Scale?  How  many 
pounds  now  make  a  ton  ?  How  many  formerly  ?  What  are  the  different  tons 
called  ?  For  what  is  the  long  ton  now  used  ?  One  pound  Avoirdupois  equals 
how  many  grains  Troy  ? 


82 


REDUCTION, 


Ex.  1.  Eeduce  2t.  6cwt. 
Iqr.  231b.  to  pounds. 

OPERATION. 

2t.  6cwt.  Iqr.  231b. 
20 

4  6  cwt. 
4 

1  8  5  qr. 
25 


925 
370 

2  31b. 


Ex.  2.  In  46481b.  how  many 
tons,  etc.  ? 

OPERATION. 

2  5  )  4  6  4  8  lb. 


4  )  1  8  0  qr.    +    231b. 
2  0)  4  6  cwt.  +    Iqr 
2t.      -f   6cwt. 


Ans.  2t.  6cwt.  Iqr.  231b. 


4  6  4  8  lb.,  Ans. 

Note  3.  Instead  of  mentally  adding  the  numbers  of  the  lower 
denominations,  as  in  preceding  examples  and  as  is  done  with  the 
6cwt.  and  Iqr.  in  Ex.  1,  the  pupil  may,  if  he  chooses,  write  the 
numbers  under  the  partial  products,  and  then  add,  as  is  done  with 
the  231b.  in  this  Example. 

3.  Eeduce  6t.  7cwt.  3qr.  211b.  looz.  7dr.  to  drams. 

4.  Eeduce  4t.  2qr.  15oz.  to  drams. 

5.  Eeduce  147683dr.  to  higher  denominations. 

6.  Eeduce  1860861  dr.  to  ounces,  pounds,  quarters,  etc. 

7.  If  a  cow  eats  161b.  of  hay  in  1  day,  how  many  tons 
will  she  eat  in  365  days? 

8.  In  7t.  16cwt.  3qr.  51b.  net  weight,  how  mz.iiy gross  toils'^ 

CLOTH  MEASUEE. 
101.      Cloth  Measure  is  used  in  measuring  cloths,  rib- 
bons, braids,  etc. 


101.    Explain  Ex.  1.    Explaia  Ex.  2.    101.    For  what  is  Cloth  MeaRTire 
used  t    Tabic  ?    Scale » 


21  Inclies  (in.) 
4  Nails 
4  Quarters 


REDUCTION. 

TABLE. 

make 


TJirlVia^ 


1  Nail.  ^ 
1  Quarter 
1  Yard, 


O?'^ 


na. 


qr. 
yd. 


yd. 


qr. 

1         = 
-        4        = 


na. 
1 
4 


in. 
9 


16         =z       36 


Scale.     Descending,  4,  4,  21 ;  Ascending,  2J,  4,  4. 

1.  In  6yd.  2qr.  3na.  how  many  nails? 

2.  In  107  nails  how  many  quarters,  etc.  ? 

3.  Reduce  18yd.  Iqr.  2na.  to  nails. 

4.  Reduce  47yd.  3qr.  Ina,  to  nails. 

5.  Reduce  783  nails  to  quarters,  etc. 

6.  Reduce  549  nails  to  higher  denominations. 

7.  If  2yd.  Iqr.  of  ribbon  are  used  in  trimming  1  bonnet, 
how  many  yards  will  be  used  in  trimming  5  bonnets  ? 

8.  If  2yd.  Iqr.  3na.  of  cloth  are  used  in  making  1  coat, 
how  many  yards  will  be  used  in  making  16  coats?   Aiis.  19. 

9.  How  many  dresses  can  be  made  from  117yd.  2qr.  of 
silk,  if  each  dress  requires  14yd.  2qr.  3na.  Ans.  8. 

10.      What  cost  18yd.  3qr.  of  velvet,  at  $2  per  quarter? 

LONG  MEASURE. 

103.  Long  Measure  is  used  in  measuring  distances; 
as,  for  example,  the  length  of  a  line,  or  the  length,  breadth, 
height,  or  depth  of  any  object. 


lOa    For  what  is  Long  Measure  used  ?    Table  ?    Scale  ? 


04                                          REDUCTION. 

TABLE. 

12  Inches  (in.)           make 

1  Foot, 

ft 

3  Feet 

1  Yard, 

yd. 

5iYardsor  le^Feet '' 

1  Rod, 

rd. 

40  Kods 

1  Furlong, 

fur. 

8  Furlongs                   " 

1  Mile, 

m. 

69^  Statute  miles,  nearly  «' 

1  Degree  OB 

iCirc.  of  the  Earth,  V 

360  Degrees 

1  Circumference, 

circ. 

ft. 

in. 

yd. 

1     = 

12 

rd. 

1   = 

3     — 

36 

fur.              1     r= 

H  = 

m  = 

198 

m.           1     r=    40    = 

220     =z 

660     = 

7920 

1  =:  8  =  320 


1760  =  5280  =  63360 


Scale.  Descending,  360,  69j,  8,  40,  5J,  3,  12 
ing,  12,  3,  5i,  40,  8,  69^,  360. 


Ascend- 


Note  1.  The  earth  not  being  an  exact  sphere,  the  distance  round 
it  in  difterent  directions  is  not  exactly  the  same.  By  the  most  exact 
measurements  made,  a  degree  is  a  little  less  than  69^  miles. 

Note  2.  Besides  the  numbers  given  in  the  table,  there  are  vari- 
ous other  measures  of  length ;  as,  3  barleycorns  make  1  inch,  4 
inches  1  hand,  9  inches  1  span,  3  feet  1  space,  6  feet  1  fathom,  3  geo- 
graphic miles  1  league,  60  geographic  miles  1  degree,  etc. 

1.  How  many  rods  in  5m.  6 fur.  37rd.  ? 

2.  Reduce  1877  rods  to  higher  denominations. 

3.  Reduce  3659  rods  to  higher  denominations. 

4.  In  301  furlongs  how  many  miles? 

5.  In  5yd.  1ft.  9 in.  how  many  inches  ? 

6.  In  197  inches  how  many  feet,  etc.  ? 

7.  The  distance  through  the  earth  is  about  7912  miles;  how 
many  rods  is  it  ? 


REDUCTION.  85 

8.  The  distance  round  the  earth  is  about  8000000  rods  J 
how  many  miles  is  it  ?  Ans.  25000. 

9.  The  distance  from  the  earth  to  the   moon   is  about 
240000  miles;  how  many  rods  is  it  ? 

10.     The  distance  from   the   earth   to   the   sun    is   about 
30400000000  rods ;  how  many  miles  is  it? 

CHAIN  MEASURE. 
103.     Chai?^  Measure  is  used  by  engineers  and  surveyors 
in  measuring  roads,  canals,  boundaries  of  fields,  etc. 


TABLE. 

7-1^0-0  Inches  (in) 

make 

1  Link, 

li. 

25 

Links 

ii 

1  Rod,  Perch 

»or 

Pole,  rd. 

4 

Rods 

It 

1  Chain, 

ch. 

10 

Chains 

« 

1  Furlong, 

fur. 

8 

Furlongs 

• 

(( 

1  Mile, 

li. 
rd.                    1 

m, 
in. 

ch. 

1      =:           25 

z= 

198 

fur. 

1 

z= 

4     =       100 

r= 

792 

m. 

1     = 

10 



40     =     1000 

— 

7920 

1      : 

=     S     = 

80 

izr       J 

320     =     8000 

= 

63360 

Scale.     Descending, 

8,  10 

,  4,  25,  7^%^j;  Ascending  7x9/^, 

25,4. 

10,  8. 

Note.     To  measure  roads,  etc.,  engineers  often  use  a  chain  100 
feet  long. 

1.  Reduce  3m.  4fur.  8ch.  2rd.  20li.  to  links. 

2.  Reduce  28870  links  to  higher  denominations. 

3.  Reduce  5  m.  7fur.  3ch.  to  links. 

103    A  degree  upon  the  earth,  how  long  ?  What  other  measures  of  length  ? 
103    For  what  is  Chain  Measure  used  ?    Table  ?    Scale  ?    Note  ? 


SQ 


REDUCTION. 


4.     Eeduce  4m.  Sch.  221i.  to  links. 

6.     Eeduce  35G47  links  to  higher  denominations. 

6.  The  distance  from  Boston  to  Andover  is  about  1 84000 
links ;  how  many  miles  is  it  ? 

7.  The  distance  round  a  field  is  5 far.  7ch.  3rd. ;  what  will 
it  cost  to  fence  this  field  at  $  2  per  rod  ? 

SQUARE  MEASURE. 
104:*     A  Surface  is  that  which  has  length  and  breadth 
but  no  thickness. 

10^.     A  four-sided  figure  having  all  its  comers  or  angles, 
equal  to  each  other,  as  A  B  C  D,  Fig.  1,  is  called  a  Rectangle^ 
A  Fig.  1.  B    .       A      Fm.  2.        B 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

.13 

14 

15 

B 


D 


106.  A  Rectangle  whose  sides  are  all  equal  to  each  other, 
as  A  B  C  D,  Fig.  2,  is  called  a  Square.  The  small  checks  in 
Fig.  1  are  squares. 

107.  The  manner  of  finding  the  area  or  measure  of  any 
rectangle,  as,  for  example,  Fig.  1,  may  be  understood  by  the 
following  explanation:  —  Let  A  B  represent  (on  a  reduced 
scale)  a  line  five  feet  long ;  then,  evidently,  if  we  pass  from 
A  to  e,  a  distance  of  1  foot,  and  draw  the  line  e/ the  figure  A 
B/e  will  contain  5  square  feet,  that  is  5  X  1  square  feet.  So,  in 


104:      What  is  a  Surface?      105.      A  Rectangle? 
107.    How  is  the  area  of  a  Rectangle  found  ? 


106.      A  Square? 


REDUCTION. 


87 


like  manner,  A  B  A^  will  contain  10,  or  5  X  2  square  feet, 
and  A  B  C  D  will  contain  15,  or  5  X  3  square  feet.  Hence 
we  multiply  together  the  numhtrs  representing  the  length  and 
breadth  of  a  rectangle  to  find  its  area. 

108.     Reversing  the  process  in  Art.  107,  r 

The  area  of  a  rectangle  divided  by  its  length  will  give  its 

breadth,  and  the  area   divided  by   the   breadth   will  give  the 

length  ;  thus,  in  Fig.  1,   15  -j-  5  zzz  3,  the  breadth,  and  15  -^ 

3  r=  5,  the  length, 

100.     Square  Measure  is  used  for  measuring  surfaces. 

TABLE. 

144     Square  laches  (sq.  in.)     make 
9     Square  Feet  " 


1   Square  Foot,    sq.  ft 
1   Square  Yard,  sq.  yd. 


30^  Square  Yards  or  | 
272i  Square  Fe^t         j 

<< 

1 

Square  Rod, 

eq.  rd. 

40     Square  Eods 

<< 

1 

Rood. 

r. 

4     Roods 

ti 

1 

Acre, 

a. 

640     Acres 

n 

1 

Square  Mile, 

sq.  m. 

(a)  Also  in  Chain  Measure, 

10000  Square  Links  or") 
16  Square  Rods        ) 

make 

1 

Square  Chair 

I,  sq.  ch. 

10  Square  Chains 

<« 

1 

Acre, 

sq.  ft. 

a. 
sq.  in. 

sq.  yd. 

1  = 

144 

sq.  rd. 

1 

= 

9  = 

1296 

r.                 1  = 

30i 

= 

272^  = 

39204 

a.              1  =         40  = 

1210: 

= 

10890  = 

1568160 

sq.  m.      1  =        4  =        160  = 

4840: 

= 

43560  = 

6272640 

1  =  640  =  2560  =  102400  = 

3097G00  : 

=  27878400  =  4014489600 

108.  How  is  the  breadth  of  a  rectangle  found  when  the  area  and  length  are 
known  ?    How  the  length  when  the  area  and  breadth  are  known  ? 

109.  For  what  is  Square  Measure  used  ?    Table  ?    Scale  ?    Table  in  Chaui 
Measure?    Note  1?    Note 2? 


8S  REDUCTION. 

Scale.  Descending,  640,  4,  40,  30^,  9,  144  ;  Ascending, 
144,  9,  30i  40,  4,640.  "" 

Note  1.  In  measuring  land,  surveyors  use  a  4-rod  chain  com- 
posed of  100  links.     Sometimes  the  half-chain  of  50  links  is  used. 

Note  2.    The  272^ before  feet  in  the  table  is  not  a  part  of  the  scale. 

1.  Reduce  3sq.  m.  325a.  2r.  37sq.  rd.  to  square  rods. 

2.  In  359317  square  rods  how  many  square  miles,  acres, 
roods,  and  rods  ? 

3.  Reduce  30sq.  yd.  Isq.  ft.  127sq.  in.  to  square  inches. 

4.  Reduce  39151  sq.  in.  to  higher  denominations. 

5.  How  many  square  feet  in  Fig.  1  ? 

6.  How  many  feet  round  Fig.  1  ? 

7.  Suppose  each  side  of  Fig.  2  to  be  7  rods,  what  is  the 
distance  round  it  ?     How  many  square  rods  does  it  contain  ? 

8.  How  many  square  rods  in  a  rectangular  field  that  is 
17  rods  wide  and  35  rods  long?  How  many  acres ?  How 
many  rods  round  this  field  ? 

9.  A  board  containing  45  square  feet  is  15  feet  long  ;  how 
wide  is  it? 

10.  A  flower-garden  containing  288  square  feet  is  12  feet 
wide  ;  how  long  is  it  ?  Ans.  24. 

1 1.  How  many  square  yards  of  carpeting  will  be  required 
to  carpet  a  room  that  is  18  feet  long  and  15  feet  wide  ? 

12.  At  $2  per  yard  for  carpeting  that  is  a  yard  wide, 
what  will  be  the  cost  of  carpeting  a  room  that  is  5  yards 
square  ? 

CUBIC  OR  SOLID  MEASURE. 
110.     A  Solid  or  Body  is  any  thing  which  has  length, 
breadth,  and  thickness. 

110.    What  is  a  Solid  or  Body  ? 


REDUCTION. 


89 


111.     A  solid  or  "body  bounded  by  6  rectangular  faces,  as 
Fig.  3,  is  called  a  Rectangular  Prism. 

D  Fig  3.  G 


1    i    =    i  « 

i 1 ...„ 1 

1     i     .    ' 

~  - 

— .... 





E 


113.      A    rectangular     prism  3  feet 

bounded  by  six  square  faces,  as  Fig. 
4,  is  called  a  Cube.  ^ 


Fig.  4. 


\ 


^ 
^ 


Length.  ' 
113.  To  find  the  volume  or  solid  contents  of  a  rectangu- 
lar prism,  as  Fig.  3,  first  find  the  area  of  the  top  face, 
AB  C  Dy  as  in  Art.  107;  then  going  from  Ay  B,  and  C  down- 
ward 1  foot  to  a,  h,  and  c,  and  passing  a  plane  through  a,  h, 
and  c,  we  shall  cut  ofi"  15  solid  feet,  that  is  5  X  3  X  1  solid 


111.    A  Rectangular  rrism  ? 
of  a  rectangular  prism  found  ? 


113.    A  Cube?     113.  How  is  the  volume 


90  REDUCTION. 

feet.  So,  if  a  plane  be  passed  through  d,  e,  and  /,  it  will  cut 
off  30,  or  5  X  3  X  2  solid  feet,  etc.;  that  is, 

The  continued  product  of  the  numbers  representing  the 
length,  breadth,  and  height  will  give  the  volume  or  solid  contents 
of  a  rectangular  prism  ;  thus,  in  Fig.  3,  5  X  3  X  4  =  60, 
(solid  feet,)  the  volume  or  contents. 

114:.      So,  reversing  the  process  in  Art.  113 

The  volume  divided  by  the  area  of  the  top  face  will  give  the 
height  of  the  prism  ;  the  volume  divided  by  the  area  of  one  end 
will  give  the  length  ;  and  the  volume  divided  by  the  area  of  one 
side  will  give  the  breadth  or  width;  thus,  in  Fig.  3,  60  -^-  15 
=  4,  the  depth;  60  -^  12  =  5,  the  length  ;  and  60  -^  20  = 
3,  the  breadth. 

V1.5.  Solid  or  Cubic  Measure  is  used  in  measuring 
things  which  have  length,  breadth,  and  thickness. 

TABLE. 

1728  Cubic  Inches  (c.  in.)  make    1  Cubic  Foot,   cu.  ft 

27  Cubic  Feet  "      1  Cubic  Yard,   c.  yd. 

16  Cubic  Feet  «'      1  Cord  Foot,       c.  ft 

8  Cord  Feet  or) 

128  Cubic  Feet    j 

cu.  ft  c  in. 

c.  yd.                    1  =           1728 

1         =         27  =         46656 

Scale.     Descending,  27,  1728  ;  Ascending,  1728,  27. 
Note  1.     The  numbers  after  27,  in  this  table,  do  not  belong  to  the 
scale. 

114.  How  the  depth,  when  the  volume  and  area  of  the  top  face  are  known  ? 
How  the  length,  when  the  volume  and  area  of  one  end  are  known  ?  How  the 
breadth,  when  the  volume  and  area  of  one  side  are  known  ?  115.  For  what 
is  Solid  Measure  used  ?    Table?    Scale?    Notel? 


1  Cord,  .  c. 


REDUCTION.  91 

Fig.  6. 

Note  2.  A  pile  of  wood,  Fig.  5,  that  is 
8  feet  long,  4  feet  wide,  and  4  feet  high, 
measures  a  cord,  and  one  foot  in  length  of 
such  a  pile  is  a  cord  foot. 

Note  3.  A  Perch  of  building  stone  or  masonry  contains  24^ 
cubic  feet.  A  pile  16i  feet  long,  Ih  feet  wide,  and  1  foot  high 
measures  a  perch. 

Note  4.  Transportation  companies  often  estimate  freight,  es- 
pecially of  light  articles,  by  the  space  occupied,  rather  than  by  the 
actual  weight.  In  this  estimate,  from  25  or  30  to  150  or  175  cubic 
feet  are  called  a  ton.  This  is  called  arbitrary  weight,  and  it  varies 
with  different  transportation  companies. 

1.  How  many  cubic  inches  in  33c.  yd.  24cu.  ft.  1635c.  in.  ? 

2.  Reduce  1582755c.  in.  to  higher  denominations. 

3.  Reduce  15c.  yd.  18cu.  ft.   1727c.  in.  to  cubic  inches. 

4.  In  5  c.  6c.  feet,  9  cubic  feet,  125  c.  in.  how  niary  cubic 
inches  ? 

5.  If  40  cu.  ft,  make  one  ton,  how  many  tons,  cubic  feet, 
etc.,  in  347859  cubic  inches? 

6.  How  many  cubic  feet  are  there  in  Fig.  3  ?  How  many 
square  feet  in  the  top  face  of  Fig.  3  ?  How  many  in  the  front 
side  ?  How  many  in  the  right-hand  end  ?  How  many  in  the 
whole  surface  of  Fig.  3. 

7.  How  many  cubic  feet  in  a  cubical  box  whose  edges  are 
2  feet  in  length?  How  many  cubic  inches?  How  many 
square  feet  in  its  surface  ? 

8.  How  many  cords  of  wood  in  a  pile  that  is  32  ft.  long,  4 
ft.  wide,  and  6  ft.  high?  How  many  cord  feet?  Cubic  feet? 
Cubic  inches? 

9.  A  rectangular  block  of  marble  which  contains  88  cubic 
feet,  is  1 1  feet  long  and  4  feet  wide ;  how  thick  is  it  ? 

115.    Note  2?    Notes?    Note  4? 


92  REDUCTION. 

10.  A  grain-bin  which  holds  30  cuhic  feet  of  grain  is  3  feet 
deep  and  2  feet  wide  ;  how  long  is  it?  Ans.,  5  feet 

11.  My  cistern  is  18  feet  long,  15  feet  wide,  and  10  feet 
deep.  By  a  pipe  6  cubic  feet  of  water  enter  every  minute ; 
in  how  many  minutes  will  the  cistern  be  filled  ? 

LIQUID  MEASURE. 
116.  Liquid  Measure  is  used  in  measuring  all  liquids. 


TABLE. 

4  Gills  (gi ) 

make 

1 

Pint,         pt. 

2  Pints 

ti 

1 

Quart,      qt. 

4  Quarts 

tt 

pt. 

1 

Gallon,  gal. 

qt. 

1 

=          4 

gal. 

1 

=          2 

=          8 

1            = 

4 

=:             8 

=        32 

SCALE.     Descending,  4, 

2,  4 ;  Ascend 

ing,  4,  2,  4. 

Note  1.  The  United  States  Standard  Unit  of  Liquid  Measure  is 
the  old  English  wine  gallon,  which  contains  231  cubic  inches. 

Note  2.  It  has  been  customary  to  measure  milk,  and  also  beer, 
ale  and  other  malt  liquors,  by  beer  measure,  the  gallon  containing 
282  cubic  inches,  but  this  custom  is  fast  going  out  of  use. 

Note  3.  Casks  of  various  capacities,  from  50  to  150  or  more 
gallons,  are  indiscriminately  called  hogsheads,  pipes,  butts,  tuns, 
etc.     Those  containing  from  30  to  40  gallons  are  called  barrels. 

1.  Reduce  9gal.  3qt.  Ipt.  2gi.  to  gills. 

2.  Eeduce  318  gills  to  pints,  quarts,  etc. 

3.  Eeduce  12gal.  Ipt.  to  gills. 

4.  Reduce  573  gills  to  higher  denominations. 

116.    For  what  is  Liquid  Measure  used  ?    Table  ?    Scale  ?    Note  1  ? 


REDUCTION.  93 

5.  How  many   bottles,  each   containing   3qt.  Ipt.  2gi.,  can 
be  filled  from  a  cask  which  contains  46gal.  3qt  Ipt.  ? 

6.  How  many  gallons  of  molasses  in  2i  jugs,  each  contain- 
ing Igal.  2qt.  Ipt.  ? 

DRY   MEASURE. 
117.     Dry  Measure  is  used  in  measuring  grain,  fruit, 
potatoes,  salt,  charcoal,  etc, 

TABLE. 


2  pts.  (pt.) 
8  Quarts 
4  Pecks 

make 

1 
I 
1 

Quart, .       qt 
Peck,         pk. 
Bushel,  bush. 

bush. 

pk. 

1           = 

qt. 
1 

8 

pt. 
—           2 
=         16 

1           — 

4           — 

32 

z=         64 

Scale.     Descending,  4,  8,  2  ;  Ascending,  2,  8,  4. 

Note.  The  bushel  measure  is  18^  inches  in  diameter  and  8  inches 
deep,  and  contains  a  little  less  than  2150^  solid  inches,  or  nearly  9 J 
■vrine  gallons.  Consequently  4  quarts  or  half  a  peck  of  oats  should 
measure  nearly  38  cubic  inches  more  than  a  gallon  of  wine ;  and  a 
quart  of  berries,  or  any  other  article  measured  by  Dry  Measure, 
should  contain  nearly  dh  cubic  inches  more  than  a  quart  of  wine  or 
any  other  liquid. 

1.  Reduce  3bush.  2pk.  7qt.  Ipt.  to  pints. 

2.  Reduce  239  pints  to  quarts,  pecks,  etc. 

3.  How  many  pints  in  25 bush.  Ipk.  5qts.  Ipt.  ? 

4.  How  many  pints  in  IZbush.  3qt.  ? 

116.  Note  2  ?  Note  3  ?  117.  For  what  is  Dry  Measure  used  ?  Table  ? 
Scale  ?  What  are  the  dimensions  of  the  bushel  measure  ?  How  many  cubic 
inches  does  it  contain  ?  How  many  wine  gallons  ?  How  much  ought^a  quart 
of  berries  to  exceed  a  quart  of  milk  ? 


94 


REDUCTTOX, 


5.  Reduce  759  pints  to  higher  denominations. 

6.  Reduce  8573  pints  to  higher  denominations. 

7.  What  is  the  cost  of  2bush.  3pk.  of  grass  seed,  at  $2  a 
peck? 

TIME. 

118.  Time  is  used  in  measuring  duration.  The  natural 
divisions  of  time  are  days,  months,  (moons),  seasons,  and 
years.  The  artificial  divisions  are  seconds,  minutes,  hours, 
weeks,  etc. 

TABLE. 

60  Seconds  (sec.)  make     1  Minute,  m. 

60  Minutes  *•        1  Hour,  h. 

2 1  Hours  "1  Day,  d. 

7  Days  "1  AVeek,  wk. 

4  Weeks  "         1  Lunar  Month,  1.  m. 

13  Months,  1  Day,  and  6  Hours  **         1  Julian  Year,    J.  yr, 

12  Calendar  Months  (=365  or  366  Days),  1  Civil  Year,  c.  yr. 

100  Years  make       1  Century,  C. 


1.  m. 
•  yr.  1 

1=  13^4 


wk. 

1 
=  4 


1  z= 

r=      28  = 


h. 

1  z= 

24=: 


m. 
1  — 
60  = 
1440  = 


sec. 

60 

3600 

86400 


168  =  10080  =     604800 
672  =:  40320  =  2419200 


T\2 


:52^\  =3651  —8766  =525960  =31557600 


Scale.     Descending,  4,  7,  24,  60,  60  ;  Ascending,  60,  60, 
24,  7,  4. 


118.«For  what  is  Time  used?    What  are  its  natural  divisions  ?    Artificial 
divisions  ?    Table  ? 


REDUCTION.  95 

Note  1.     The  names  of  the  seasons  and  of  the  calendar  months 
and  the  number  of  days  in  the  several  months,  are  as  follows  :  — 

Seasons.  Months.  No.  of  Days. 

rrri--.-_         (  1st.  January 31 

•        }  2d.  February 28,  in  leap  year  29 

(  3d.  March 31 

Spring.         )  4th.  April 30 

(  5th.  May 31 

(  6th.  June 30 

Summer.      ]  7th.  July 31 

<  8th.  August 31 

t  9th.  September 30 

Autumn.      ]   loth.    October 31 

<  nth.    November 30 

Winter.  12th.    December 31 

Note  2.     The  number  of  days  in  each  month  may  be  easily  re- 
membered by  committing  the  following  lines  :  — 

Thirty  days  hath  September, 
April,  June,  and  November  j 
All  the  rest  have  thirty-one, 
Save  the  second  month  alone, 
Which  has  just  eiglit  and  a  score 
Till  leap  year  gives  it  one  more. 

Note  3.     A  solar  year,  that  is,  a  year  by  the  sun,  is  very  nearly 
365  days,  5  hours,  48  minutes  and  50  seconds. 

1.  How  many  seconds  in  18]a.  27m.  30sec.  ? 

2.  Reduce  12850  seconds  to  higher  denominations. 

3.  Reduce  4d.  22li.  57m.  54sec.  to  seconds. 

4.  Reduce  9wk.  15li.  19sec.  to  seconds. 

5.  Reduce  452897  seconds  to  higher  denominations. 

6.  In  7  centuries  how  many  calendar  months  ? 

7.  Reduce  10800  calendar  months  to  centuries. 

CIRCULAR  MEASURE. 
110,   Circular  Measure  is  used  in  surveying,  naviga- 
tion, geography,  astronomy,  etc.,  for  measuring  angles,  deter- 
mining latitude,  longitude,  etc. 


118.  Scale  ?  What  are  the  names  of  the  calendar  months  ?  How  many. days 
in  each  ?  In  what  season  is  each  ?  The  number  of  each  from  the  beginning  of 
the  year  ?    Length  of  a  solar  year  ? 


y(5 

REDUCTION 
TABLE. 

60  Seconds  (60'0 

make 

1  Minute, 

1' 

60  Minutes 

(< 

1  Degree, 

1° 

30  Degrees 

it 

1  Sign, 

s. 

12  Signs,  or  360° 

<( 

1  Circumference,  circ. 

V  = 

60" 

s. 

lo   =: 

60  = 

3600 

circ.              1   zn 

30  = 

1800  =z 

108000 

I  =       12  =r 

360  z= 

21600  =z 

1296000 

Scale.     Descending, 

12,   30,   60, 

60  ;  Ascending, 

60,  60, 

30,  12. 

Note.  A  curved  line  is  a  jSgure  bounded 
by  a  curved  line,  all  parts  of  the  curve 
being  equally  distant  from  the  center  of 
the  circle. 

The  Circumference  is  the  curve  which 
bounds  the  circle.  An  Arc  is  any  portion 
of  the  circumference,  as  A  B  or  B  D. 
An  arc  equal  to  a  quarter  of  the  circum- 
ference, or  90°,  is  called  a  quadrant.  A 
Radius  is  a  line  drawn  from  the  center 
to  the  circumference,   as   C  A  or  C  B. 

A  Diameter  is  a  line  drawn  through  the  center  and  limited  by  the 

curve,  as  A  D. 

1.  In  15°  38'  29"  how  many  seconds  ? 

2.  Eeduce  78695"  to  degrees,  etc. 

3.  Keduce  2°  2T  39"  to  seconds. 

4.  In  5s.  17°  how  many  minutes? 

5.  Keduce  276892"  to  signs,  etc. 

6.  Keduce  17s.  21°    28'  3"  to  seconds. 


119.    For  what  is  Circular  Measure  used?    Table?    Scale?    What  is  a 
Circle?    Circumference?    Arc?    Quadrant?    Radius?    Diameter? 


REDUCTION.  97 

Miscellaneous  Examples  in  Eeduction. 

1.  In  7£  15s.  6d.  3qr.  how  many  farthings? 

2.  Keduce  67219qr.  to  pounds  sterling,  etc. 

3.  Eeduce  lOoz.  17dwt.  15gr.  to  grains. 

4.  Change  27&19gr.  to  pounds,  etc. 

5.  In  7oz.  5dr.  2sc.  12gr.  of  opium,  how  many  grains? 

6.  Reduce  17  tons  16cwt.  3qr.  to  quarters. 

7.  Change  627243oz.  to  tons,  etc. 

8.  In  7yd.  3qr.  2na.  lin.  how  many  inches  ? 

9.  Keduce  742  inches  to  yards,  etc. 

10.  Change  5fur.  13rd.  7ft.  lOin.  to  inches. 

11.  Eeduce  273894  inches  to  miles,  etc. 

1 2.  In  27  fathoms,  how  many  inches  ? 

13.  John   Smith's   horse   is    15    hands  high  ;  how  many 
inches  high  is  he  ? 

14.  In  7m.  3fur.  7ch.  2rd.  how  many  links  ? 

15.  Eeduce  3a.  2r.  27sq.  rd.  127sq.  ft.  126sq.  in.  to  square 
inches. 

IG.  How  many  cu.  in.  in  17  cords? 

17.  Reduce  76493c.  in.  to  cords,  etc. 

18.  IIow  many  gills  in  27gal.  3qt.  Ipt.  3gi.  ? 

19.  Eeduce  643gi.  to  gallons,  etc. 

20.  Change  46bu.  3pk.  6qt.  Ipt  to  pints. 

21.  In  874qt  how  many  bushels  ? 

22.  Eeduce  17h.  56m.  433ec.  to  seconds. 

23.  Eeduce  178cwi  2qr.  101b.  to  ounces. 

24.  Eeduce  10yd.  2na.  to  nails. 

25.  Eeduce  726890  inches  to  miles. 


98  DEFINITIONS  AND  GENERAL  PRINCIPLES. 


DEFINITIONS  AND  GENERAL  PRINCIPLES. 

130.      All  numbers  are  even  or  odd. 

An  Even  Number  is  a  number  that  is  divisible  by  2  ;  as 

2,  4,  8,  12. 

An  Odd  Number  is  a  number  that  is  not  divisible  by  2  ;  as 
1,  3,  5,  11,  19. 

ISl.     AH  numbers  aro  prime  or  composite. 

A  Prime  Number  is  a  number  that  is  divisible  by  no 
whole  number  except  itself  and  one  ;  as  1,  2,  3,  5,  7,  11,  19. 

A  Composite  Number  is  a  number  that  is  divisible  by  other 
numbers  besides  itself  and  1  ;  thus,  6  is  composite,  because 
it  is  divisible  by  2  and  by  3  ;  12  is  composite,  because  it  is 
divisible  by  2,  3,  4,  and  6  ;  25  is  composite,  because  it  is 
divisible  by  5  and  5. 

Factoring  Numbers. 

133.  The  Factors  of  a  number  are  those  numbers 
whose  continued  product  is  the  number ;  thus,  3  and  7  are 
the  factors  of  21  ;  3  and  6,  or  3,  3,  and  2  are  the  factors  of 
18;  etc. 

The  prime  factors  of  a  number  are  those  prime  numbers 
whose  continued  product  is  the  number ;  thus,  the  prime  fac- 
tors of  12  are  2,  2,  and  3  ;  the  prime  factors  of  36  are  2,  2, 

3,  and  3  ;  etc. 

NoTK.     Since  1,  as  a  factor  is  useless,  it  is  not  here  enumerated. 

130.  What  is  an  Even  Number?  An  Odd  Number?  131.  A  Prime 
Number?  What  is  the  only  even  prime  number?  What  is  a  Composite 
Number  ? 

133.  What  are  the  Factors  of  a  number  ?  Wliat  are  the  prime  factors  of 
a  number  ? 


DEFINITIONS  AND  GENERAL  PKINCIPLES. 


99 


TABLE  OF  PRIME  NUMBERS  FROM  1  TO  997. 


1 

|41 

101 

167 

239 

313 

397 

467 

569 

643 

733 

823  911 

2 

43 

103 

173 

241 

317 

401 

479 

571 

647 

739 

827 

919 

3 

47 

107 

179 

251 

331 

409 

487 

577 

653 

743 

829 

929 

5 

53 

109 

181 

257 

337 

419 

491 

587 

659 

751 

839 

937 

'  7 

59 

113 

191 

263 

347 

421 

499 

593 

661 

7Ci7 

853 

941 

11 

61 

127 

193 

269 

349 

431 

503 

599 

673 

761 

857 

947 

13 

67 

131 

197 

271 

353 

433 

509 

601 

677 

769 

859 

953 

17 

71 

137 

199 

277 

359 

439 

521 

607 

683 

773 

863 

967 

19 

73 

139 

211 

281 

367 

443 

523 

613 

691 

787 

877 

971 

23 

79 

149 

223 

283 

373 

449 

541 

617 

701 

797 

881 

977 

29 

83 

151 

227 

293 

379 

457 

547 

619 

709 

809 

883 

983 

31 

89 

157 

229 

307 

383 

461 

557 

631 

719 

811 

887 

991 

37 

97 

163 

233 

311 

389 

463 

563 

641 

727 

821 

907 

997 

1^3.  To  resolve  or  separate  a  number  into  its  prime 
factors  we  have  the  following : 

HuLE.  Divide  the  given  number  hy  -any  prime  number 
greater  than  one^  that  will  divide  it ;  divide  the  quotient  by 
any  prime  number  greater  than  one  that  will  divide  it,  and  so 
on  till  the  quotient  is  prime.  The  several  divisors  and  last 
quotient  will  be  the  prime  factors  sought. 

1.     What  are  the  prime  factors  of  5768  ? 

OPERATION. 

2)5768 


2)2884 
2)  1442 
7)     721 


10  3  Ans.  2,  2,  2,  7,  1  0  3. 


123.    Rule  for  resolving  a  number  into  its  prime  factors ' 


TOO        DEFINITIONS  AND  GENERAL  PRINCIPLES. 

2.  Eesolve  680  into  its  prime  factors. 

Ans.  2,  2,  2,  5,  and  17. 

3.  Eesolve  846  into  its  prime  factors. 

Ans.  2,  3,  3,  47. 

4.  What  are  the  prime  factors  of  200  ? 

Ans.  2,  2,  2,  5,  5. 

5.  Eesolve  984  into  its  prime  factors. 

Ans.  2,  2,  2,  3,  41. 
Greatest  Common  Divisor. 
134.     A  Common  Divisor  of  two  or  more  numbers  is  any 
number  that  will  divide  each  of  them  without  remainder  ;  thus 
3  is  a  common  divisor  of  12,  18,  and  30. 

13^.  The  Greatest  Common  Divisor  of  two  or  more 
numbers  is  the  greatest  number  that  will  divide  each  of  them 
without  remainder  ;  thus,  6  is  the  greatest  common  divisor  of 
12,  18,  and  30. 

Note.  A  divisor  of  a  number  is  often  called  a  measure  of  the 
number,  also  an  aliquot  part  of  the  number. 

ISO.  To  find  the  greatest  common  divisor  vre  have 
the  follovring : 

EuLE  1.  Divide  the  greater  of  two  numbers  by  the  less, 
andf  if  there  be  a  remainder,  divide  the  divisor  by  the  remain- 
der, and  continue  dividing  the  last  divisor  by  the  last  remainder 
until  nothing  remains  ;  the  last  divisor  is  the  greatest  common 
divisor  of  the  two  numbers.      Ur, 

EuLE  2.  If  more  than  two  numbers  are  g'ven^  find  the 
greatest  divisor  of  two  of  them,  then  of  this  divisor  and  a 
third  number,  and  so  on  until  all  the  numbers  have  b  en 
taken  ;  tJie  last  divisor  will  be  the  divisor  sought. 

134.  What  is  a  common  divisor  ?  135.  What  is  the  greatest  common 
divisor  ? 

136.  Rule  for  finding  the  greatest  common  divisor  of  two  numbers  ? 
Second  rule  for  finding  greatest  common  divisor  of  more  than  two  numbers  ? 


I 


DEFINITIONS  AND  GENERAL  PRINCIPLES.  101 

1.  What  is  the  greatest  com-  2.  What  is  the  greatest  com- 
tnon  divisor  of  16  and  44  ?        mon  divisor  of  8,  12,  28  ? 

OPERATION.  OPERATION. 

'16)44(2  8)12(1 

32  8 

12)16(1  4)8(2 

12  8 

Ans.,  4)12(3  Ans.,  4  )  2  8  (7 

12  28 

3.  Find  the  greatest  common  divisor  of  9,  12,  18,  and 
24.  Ans.  3. 

4.  What  is  the  greatest  common  divisor  of  24,  40,  68  ? 

Ans.  4. 

5.  What  is  the  greatest  common  divisor  of  144,  17,  and 
1728? 

6.  What  is  the  greatest  common  divisor  of  72,  45,   999  ? 

7.  What  is  the  greatest  common  divisor  of  1825,  640,  60? 

Least  Common  Multiple. 

137.  A  Multiple  of  a  number  is  any  number  which 
is  divisible  by  that  number  ;  thus,  15  is  a  multiple  of  5  and 
also  of  3  ;  21  is  a  multiple  of  7  and  of  3. 

138.  A  Common  Multiple  of  two  or  more  numbers,  is 
any  number  which  is  divisible  by  each  of  the  given  numbers ; 
thus,  48  is  a  common  multiple  of  4,  6,  and  8. 

139.  The  Least  Common  Multiple  of  two  or  more  num- 
bers, is  the  least  number  that  is  divisible  by  each  of  the  given 
numbers ;  thus,  24  is  the  least  common  multiple  of  4,  6,  and  8. 

VZ7.    What  is  a  Multiple  of  a  number? 

Ii28.    A  Common  Multiple  of  two  or  more  numbers  ? 

1*9.    The  Least  Common  Multiple? 


102        DEFINITIONS   AND   GENERAL   PRINCIPLES. 

130.  To  find  the  least  common  multiple  of  two  or 
more  numbers  : 

Rule.  Having  set  the  given  numbers  in  a  line,  divide  hj 
any  prime  number  that  will  divide  two  or  more  of  them,  and  set 
the  quotients  and  undivided  numbers  in  a  line  beneath  ;  fro- 
ceed  with  this  line  as  with  the  first,  and  so  continue  until  no 
two  of  the  numbers  can  be  divided  by  any  number  greater  than 
one  ;  the  continued  product  of  the  divisors  and  numbers  in  the 
last  line  will  be  the  multiple  sought, 

Ex.  1..  What  i3  the  least  common  multiple  of  6,  8,  12, 
16,  18,  24? 

OPERATION. 

2)  6,  8,  12,  1  6,  18,  2  4 


2)3,  4, 

6, 

8, 

9,  12 

2)3,  2, 

3, 

4, 

9,      6 

3)3,  1, 

3, 

2, 

9,      3 

1,  1,      1,      2,      3,      1 
2X2X2X3X2X3  =  144,  Ans. 

2.  Find  the  least  common  multiple  of  5,  10,  12,  15,  20, 
24.  120,  Ans. 

3.  rind  the  least  common  multiple  of  7,  8,  12, 14, 16,  21. 

Ans.  336. 

4.  Find  the  least  common  multiple  of  24,  72,  18,  48. 

5.  Find  the  least  common  multiple  of  10,  15,  24,  18,  32. 

6.  Find  the  least  common  multiple  of  21,  7,  36,  42,  84, 
Vo. 

130.    Rule  for  fiading  the  Least  Common  Multiple  ? 


coMMOx   rr.ACTioxs.  103 

COMMON     FEACTIONS. 

ISl.  If  a  single  thing  (an  apple,  for  instance,)  is  divided 
into  two  equal  parts,  one  of  these  parts  is  called  one  half, 
(written  i)  ;  if  divided  into  three  equal  parts,  one  of  these 
parts  is  called  one  third  (J). 

€>  «  # 

Halves.  Thirds.  Fourths. 

And  so,  if  we  divide  a  unit  or  single  thing  into  four,  five, 
six,  etc.  equal  parts,  one  of  these  parts  is  called  one  fourth  (i), 
one  fifth  (I),  one  sixth  (^),  etc. 

ONE    UNIT. 


1 

i 

1 

i 

1 

1 

i 

i 

1 

1 

i 

i 

1 

1 

1- 

i 

± 

-+- 

J_ 

-+- 

i 

-+- 

i 

-i- 

i 

_^ 

J_ 

-+- 

J_ 

— 1 

H 

13^.  A  Fraction  is  an  expression  representing  one  or 
more  of  the  equal  parts  of  a  unit. 

133.  A  Common  or  Vulgar  Fraction  is  expressed  by 
two  numbers,  one  above  and  the  other  below  a  line  ;  thus  ^ 
(one  half),  f  (two  fifths),  &c 

(a)  The  number  above  the  line  is  called  the  Numerator, 
and  the  number  below  the  line  is  called  the  Denominator. 

(b)  The  Denominator  shows  into  how  many  parts  the 
unit  is  divided,  and  gives  the  name  to  the  fraction. 

133,  VThat  is  a  Fraction?  133,  A  Common  Fraction?  (a)  Where  do 
we  write  the  numerator?  Denominator?  (b)  What  does  the  denominator 
show  ?    (c)    What  the  numerator  ?    (d)    What  are  both  called  ? 


104  COMMON    FRACTIONS. 

(o)  The  Numerator  shows  how  many  of  those  parts  are 
taken  or  expressed  by  the  fraction. 

(d)  The  numerator  and  denominator  are  called  the  terms 
of  the  fraction. 

Write  the  following  fractions :  three  fourths,  two  thirds, 
seven  eighths,  nine  tenths,  seven  elevenths,  eight  fifteenths. 

Bead  the  following  fractions :  ^,  |,  |,  f ,  f ,  ^j,  |f ,  ij. 

134:.  A  Simple  Fraction  has  but  one  numerator  and 
one  denominator  ;  as  |,  f,  ^-. 

135.  A  Compound  Fraction  is  a  fraction  of  a  fraction ; 
as  S  of  f ,  f  of  -^%. 

136.  A  Proper  Fraction  is  one  whose  numerator  is  less 
than  the  denominator  ;  as  f ,  |,  f . 

137.  An  Improper  Fraction  is  one  whose  numerator 
equals  or  exceeds  the  denominator ;  as  |,  ^,  f ,  ^-. 

138.  A  Mixed  Number  is  a  whole  number  and  a  frac- 
tion united ;  as,  7^,  5|,  27f. 

139.  The  terms  of  a  fraction  sustain  to  each  other  the 
relation  of  dividend  and  divisor,  the  numerator  answer- 
insT  to  the  dividend  and  the  denominator  to  the  divisor. 

That  is,  a  fraction  may  be  regarded  as  an  expression  of 
division.     Hence, 

The  VALUE  of  a  fraction  is  the  quotient  of  the  numerator 
divided  by  the  denominator,  asf  =  9-i-3  =  3. 

It  follows  from  this  that  the  General  Principles  of 
Division  (Arts.  85,  86,  and  87)  apply  to  all  fractions. 


134,  What  is  a  Simple  Fraction ?  135.  Compound?  136.  Proper? 
137.    Improper  ?     138.    What  is  a  mixed  number  ? 

139.  What  relation  do  the  terms  of  a  fraction  sustain  to  each  other? 
Wliich  term  answers  to  the  dividend  ?  Which  to  the  divisor  ?  How  may  a 
fraction  be  regarded  ?  To  what  is  the  value  of  a  fraction  equivalent  f  What 
principles  before  stated  apply  to  fractions  ? 


COMMON    FRACTIONS.  105 

1 .  Multiplying  the  numerator ^  if  the  denominator  remains 
unaltered^  m,ultiplies  the  value  of  the  fraction  hy  the  same 
number,  as  ^"^  ^  z=z^. 

2.  Dividing  the  numerator,  if  the  denominator  remains 
unaltered,  divides  the  value  of  the  fraction  hy  the  same  number, 
asi^2  =1, 

In  the  above  cases  it  will  be  seen  that  the  size  of  the  parts, 
(fourths,)  remains  the  same,  but  the  number  of  the  parts  is  in- 
creased or  diminished. 

3.  Multiplying  the  denominator,  if  the  numerator  remains 
unaltered,  divides  the  value  of  the  fraction  by  the  same  ninnber, 

««  I  X  2  =  f  • 

4.  Dividing  the  denominator,    if  the  numerator  remxiins 

unaltered,  multiplies  the  value  of  tJie  fraction  by  the  same  num- 
ber, as  f  ^  2  =  f . 

In  the  last  two  cases  it  will  be  seen  that  the  number  of 
parts  (numerators)  remains  the  same,  but  the  size  of  the 
parts  (denominators)  is  increased  or  diminished. 

5.  If  the  numerator  and  denominator  are  both  multiplied 

or  divided  by  the  same  number  the  value  of  the  fraction  is  not 

;,      ,        2X2        4      2-^2        1 
altered,  as  -  ..  c.  z=z -or ~    ;    «  =  — 

4X2        y      4—2        2 

Hence,  the  following  general  law  in  regard  to  Fractions 
may  be  stated. 

That  any  change  in  the  numerator  causes  a  like  change  in 
the  t^alice  of  the  fraction  ;  and  any  change  in  the  denominator 
causes  an  opposite  change  in  the  value  of  the  fraction, 

LTpon  these  principles  all  the  following  operations  upon 
fractions  depend. 

139.  Give  the  Ist  principle  and  illustrate  it.  The  2d  principle.  The  3d 
principle.  The  4th  principle.  The  5th  principle.  "What  general  law  is 
given  ? 


106  COMMON    FRACTIONS. 

Case   1. 
14:0.     To  reduce  a  mixed  number  to  an  improper 
fraction. 

Ex.  1.     In  7f  how  many  fifths  ? 

17  3  '             In  2l  unit  there  are  five  fifths ;  and  in 

g^  seven   units   there    are  seven  times  five 

fifths,    or   35    fifths,    which  with  the  3 

3  3    .  fifths  in  the  example  =z  38  fifths  =  ^^-. 

EuLB.  Multiply/  the  whole  number  by  the  denominator  of 
the  fraction  ;  to  the  'product  add  the  numerator  ^  and  under  the 
sum  write  the  denominator, 

2.  Eeduce  1 7f  to  an  improper  fraction.  Ans.  ^K 

3.  Eeduce  26 14-  to  an  improper  fraction.       Ans.  ^^, 

4.  Eeduce  43f  to  an  improper  fraction.  Ans.  ^i. 

5.  Eeduce  56 §  to  an  improper  fraction.  Ans.  ^^, 

6.  Eeduce  85^j  to  an  improper  fraction.      Ans.  -^f^ 

7.  In  19y^j  how  many  fourteenths? 

8.  How  many  seventeenths  in  38|f  ?  Ans.  ^^ 

9.  Eeduce  49^^  to  an  improper  fraction. 

Note.  To  reduce  a  whole  number  to  a  fraction  having  any  given 
denominator,  multiply  the  whole  number  by  the  proposed  denomi- 
nator, and  under  the  product  write  the  denominator. 

Case  2. 

141  •     To  reduce  an  improper  fraction  to  a  vrhole  or 

mixed  number. 

Ex.  1.     How  many  units  in  ^-  ? 

In   one   unit    there   are   four 

fourths,  and  in  seventeen  fourths 
JLT.  —  17    •   4  —  4-1- 
^  •  there  are  as  many  units  as  four 

is  contained  times  in  seventeen. 

140.  Explain  the  operation  in  Case  1.  Rule  for  reducing  a  mixed  number 
to  an  improper  fraction  ? 

141.  Bule  for  reducing  an  improper  fraction  to  a  whole  or  mixed  number  ? 


COMMON     FRACTIONS.  107 

Rule.     Divide  the  numerator  by  the  denominator ;  if  there 

is  any  remainder,  place  it  over  the  divisor,  and  annex  the 
fraction  so  formed  to  the  quotient. 

2.  Reduce  4^^  to  a  mixed  number.  Ans,  2f . 

3.  Reduce  ^  to  a  mixed  number.  Ans.  5f . 

4.  Reduce  ^f  to  a  mixed  number.  Ans.  ^^. 

5.  Reduce  \^  to  a  mixed  number. 

6.  Reduce  ^/-  to  a  mixed  number.  Ans.  9^|. 

7.  Reduce  ^^-  to  a  mixed  number.  Ans.  lO^f . 

8.  Reduce  W-  ^^  *  mixed  number. 

9.  Reduce  ^^  to  a  whole  number.  Ans.  7. 
10.     Reduce  ^^'  to  a  whole  number. 

Note.  The  denominator  of  a  fraction  being  a  divisor,  it  follows 
that  whenever  the  denominator  exactly  measures  the  numerator, 
the  quotient  will  be  a  whole  number.     (See  Exs.  9  and  10.) 

Case  3. 
14^.     To  reduce  a  fraction  to  its  lowest  terms. 
Ex.  1.     Reduce  f|  to  its  lowest  terms. 

1st  OPERATION  ^^^^  ^-       ^*^^'   ^^^^   ^^^   ^y  ""''y  /«^' 

2)2^ tor  common  to  them,  then  divide  these  quo- 

\)i^~&  tients  hy  any  factor  common  to  them,  and 

^^^rC^      2  *^  proceed  till  the  quotients  are  mutually 

8 )  ^  =  f  Ans      ^^^^      ^^^^^  j3^^  .^j^^ 

2d  OPERATION.      Find  the  greatest  common  divisor,  (Art.  125,) 

l|}f|=:|  Ans.  and  by  it  divide  both  terms  of  the  fraction. 

Rule  2.      Divide  each  term  hy  their  greatest  common  divisor. 

2.  Reduce  -^|  to  its  lowest  terms.  Ans.  |. 

3.  Reduce  |-f  to  its  lowest  terms.  Ans.  ^. 

4.  Reduce  ^^f  to  its  lowest  terms. 

14t2.    Rule  for  reducing  a  fraction  to  its  lowest  terms  ?     Second  rule  for 
reducing  a  fraction  to  its  lowest  terms  ? 


108  COMMON    FRACTIONS. 

5.  Reduce  /^  to  its  lowest  terms.  Ans.  |. 

6.  Reduce  f  |^f  to  its  lowest  terms. 

7.  Reduce  ^-^-^  to  its  lowest  terms.  Ans.  ^. 

8.  Reduce  :|f  f  to  its  lowest  terms. 


Ca^e  4. 

143.     To  multiply  a  fraction  by  a  whole  number. 
Ex.  1.     Multiply  I  by  4. 
i„4.  ^^      .,.„^..  I*  is  just  as  evident  that   4   times   7 

Ist  OPERATION.  •' 

r  w  ^ 20. i     eighths  (t)  are  28  eighths  (%8-)   as  it  is 

that  4  times  7  boys  are  28  boys. 

2d  OPERATION.  If  we  divide  the  denominator  by  4  we 
5^4==^        obtain  the  same  result  as  before. 

In  the  first  operation  we  increase  the  number  of  the  parts 
four- fold,  and  in  the  second,  we  increase  the  size  or  value  of 
the  parts  four- fold  while  the  number  of  parts  remains  the 
same.     Hence  the  following 

Rule  1       Multiply  the  numerator  hy  the  whole  number.     Or, 
Rule  2       Divide  the  denominator  by  the  whole  number, 

2.  Multiply  A  ^y  3.  Ans.  ^. 

3.  Multiply  -/^  by  8.  Ans.  f  f  or  |. 

4.  Multiply  i^  by  5.  Ans.  f  f  =:  2f |. 

5.  Multiply  ^g  by  14.  Ans.  ||  or  f  =  3. 

6.  Multiply  jS^-  by  3.  Ans.  f|-  or  ^. 

7.  Multiply  ^W  by  15.  Ans.  |||. 

8.  Multiply  ^853^  by  12. 

143.    First  rule  for  multiplying  a  fraction  by  a  whole  number?      Second 
rule? 


COMMON    FRACTIONS.  109 

9.  Multiply  ^^j\  by  21. 

10.  Multiply  fl  by  117.  Ans.  ef^s. 

11.  Multiply  II  by  17. 

12.  Multiply  ^Yf  ^y  3^-  ^ns.  4-V¥- 

(a)     To  multiply  a  mixed  number  by  a  whole  number. 

13.  Multiply  ^  by  8. 

1st  OPERATION.  2 J  OPERATION. 

^  —  ^f-  ^  X  8  =  3_2  ^  ^ 

^  X  8  z=z  -i|2  _  3S|  Ans.  4  X  8  =:  32. 


32  +  6|z=38f. 


Hence  the  following 


KuLE.  Reduce  the  mixed  number  to  an  improper  fraction^ 
(^r#.  140,)  and  then  multiply.  Or^  multiply  the  fraction  and 
whole  number  separately  and  add  the  products  together. 


14. 

Multiply  6f  by  9. 

Ans.  60f 

15. 

Multiply  7  rV  ^1  26. 

Ans.  190. 

16. 

Multiply  28^\  ^y  42. 

Ans.  1186^j,. 

17 

Multiply  46fx  by  39. 

18. 

Multiply  89-t-V  by  68. 

19. 

Multiply  246J-I  by  142. 

20. 

Multiply  392/^  by  257. 

21. 

Multiply  1501-  by  27. 

as  multiplying  a  fraction  by  a  whole  number,  e.  g.  yX4=:4Xf. 

Case  5. 
14:4:.     To  divide  a  fraction  by  a  whole  number. 

143.    How  is  a  mixed  number  mxiltiplied  by  a  whole  number?    Another 
method  ? 


110  COMMON    FKAOTIONS. 

Ex.  1.     Divide  ?  by  3.  Ans.  f . 

One  third  of  6  apples  is  2  apples ;    it  is 
1st  OPERATION,     equally  clear  that  one  third  of  6  sevenths 
f  ^  3  =  f        (f )  is  2  sevenths  (f .) 

If  I  divide  f  by  1,  the  quotient  will  be 

2d  OPERATION,     f.     Now  if  I  divide  it  by  3  instead  of  1,  I 

f  X  3  =  A     obtain  a  quotient  only  one  third  as  great,  or 

^  of  f  z=  ^\.     In  this  instance  the  number 

of  parts  remains  the  same,  while  the  size  of 

the  parts  is  diminished. 

EuLE  1.     Divide  the  numerator  hy  the  whole  number,      Or^ 
EuLE  2.     Multiply  the  denominator  hy  the  whole  number, 
(Art.  139,  2nd  and  3rd.) 

2.  Divide  Jf  by  8.  Ans.  /-. 

3.  Divide  J-f  by  6.  Ans.  j^^. 

4.  Divide  \\  by  5.  Ans.  -^J. 

5.  Divide  -i-f  by  12.  Ans.  y^^. 

6.  Divide  |f  by  13.  Ans.  J3. 

7.  Divide  |f  by  7.  Ans.  j^^-. 

8.  Divide  f  f  by  14. 

9.  Divide  ^%%  by  35. 

10.  Divide  JfJ  by  42. 

Note.  If  the  dividend  be  a  mixed  number  first  reduce  it  to  an 
improper  fraction,  or  divide  the  whole  number  and  fraction  sepa- 
rately and  add  the  results. 

11.  Divide  261  by  6. 

12.  Divide  161  by  7.  Ans.  2\%, 

13.  Divide  28|  by  7. 


144.    First  rule  for  dividing  a  fraction  by  a  whole  number  ?     Second  rule  ? 
A  mixed  number  how  divided  by  a  whole  number  ? 


COMMON    FRACTIONS.  Ill 

14.  Divide  69i  by  13.  Ans.  5|. 

15.  Divide  21 1|  by  12.  Ans.  17f. 

Case  6. 
14:5.     To  multiply  a  fraction  by  a  fraction. 

Ex.  1.     Multiply  I  by  i  Ans.  fj-. 

We  first  multiply  the  fraction  |  by  7,  (Art.  143,  Eule  1.) 
and  obtain  ^^'.  Now,  as  7  is  eight  times  the  true  multiplier  I, 
the  product  is  8  times  too  large ;  and  we  obtain  the  true  prod- 
uct by  dividing  -\^~  by  8  (Art  141,  Kule  2.)  |  X  7  —  sj-, 
and  -2J-  -^  8  =z  |i.  Hence, 

Kule.  Multiply  the  numerators  together  for  a  new  nume- 
rator jand  the  denominators  for  a  new  denominator. 

2 

6  2 

2.     Multiply  izX^  -^^s.  f . 

7  p 

In  the  above  example,  we  have  the  factor  3  in  the  numerator 
of  the  f ,  and  also  in  the  denominator  of  the  f .  These  we 
reject  in  the  operation,  since  this  is  equivalent  to  dividing 
both  terms  of  the  product  by  3,  which  (Art.  139,  5th)  does 
not  alter  the  value  of  th  j  fraction,  and  obtain  the  answer  in 
its  lowest  terms.  This  process  of  cancellation  may  be  em- 
ployed advantageously  in  many  cases,  as  the  principle  is  the 
same,  as  when  applied  in  division.     (See  Art.  88.) 

17      5       17 
8.     Multiply  ^*  X  g  =  56,  Ans. 

7 

4.  Multiply  ^\  by  by  ^%,  Ans.  i|. 

5.  '  Multiply  If  by  Jf.  Ans.  -g-ff. 

6.  Multiply  tf  by  ^f.  Ans.  ^\\, 

7.  Multiply  f^  by  ff. 

145 .    Rule  for  multiplying  a  fraction  by  a  fraction  ? 


112  COMMON    FRACTIONS. 

8.  Multiply  |6-|  by  U^-. 

9.  Multiply  6 1  by  ^y-  Ans.  Ih 
Note-  1.     Eeduce  the  mixed  numbers  to  improper  fractions. 

10.  Multiply  5f  by  f . 

11.  Multiply  2^  by  §.  Ans.  9/,. 

12.  Multiply  58f  by  5^.  Ans.  298f. 

A  compound  fraction  may  be  reduced  to  a  simple  one  by  the  rule 
for  multiplying  one  fraction  by  another. 

13.  f  of  f  equals  what  ?  Ans.  t\  =  ^ 

14.  I  of  -^^  of  ^  equals  what  ? 

15.  /tj-  of  f f  of  II  equals  what  ? 

Case  7. 
140.     To  divide  a  fraction  by  a  fraction. 
Ex.  1.     Divide  f  by  f .  Ans.  |  =  1^ 

We  first  divide  |  by  2,  and  obtain 
OPERATION.  f ,  Art.  (144,  Eule  2,)  but  the  divisor 

|-^-f  =  l  X  |  =  f  used  is  3  times  too  great,  and  conse- 
quently the  quotient  f  is  only  ^  of  the 
required  quotient,  and  hence  must  be  multiplied  by  3  to  obtain 
the  correct  result. 

From  the  above  we  have  the  following 
Eule.     Invert  the  divisor,  and  then  proceed  as  in  multipli- 
cation (Art.  145). 

2.  Divide  ^  by  |A.  Ans.  |. 

3.  Divide  if  by  f.  Ans.  |f  =  l^\. 

4.  Divide  |f  by  I,  Ans.  f  ^. 

5.  Divide  4f  by  f .  Ans.  ^^  =  6-J-i. 

6.  Divide  -^^^  by  ^§. 

7.  Divide  fax  by  f 

146.    Rule  for  dividing  a  fraction  by  a  fraction  ? 


COMMOJT    FRACTIONS.  113 

8.  Divide  I  by  if  i.  Anfl.  |f|. 

9.  Divide  ff  by  ^^. 

10.     Divide  f  by  ^  of  ^.  Ans.  8. 

Note.  If  either  of  the  quantities  is  a  mixed  number  it  must  be  re- 
duced to  an  improper  fraction. 

Case  8. 
147.     To  reduce  fractions  that  have  different  denom- 
inators to  equivalent  fractions  having  a  common  denom- 
inator. 

Ex.  1.     Eeduce  |  and  f  to  equivalent  fractions  having  a 
common  denominator. 

We  multiply  both  terms  of  each  frac- 
tion  by  the   denominator  of  the  other 

5  w  ? ?J  fraction  ;  this  (Art  139,  5th)   does  not 

^       9       36  alter  the  value  of  either  fraction,  and  of 

5       4       20  necessity  it  makes  the  denominators  alike 

9  ^  4  ^^  36  as  they  are  both  the  product  of  the  two  de- 

nominators, 4  and  9. 
EuLE.     3Iultiply  both   terms  of  each  fraction  by  the  con- 
tinued product  of  the  denominators  of  all  the  other  fractions. 
Ex.  2.    Pieduce  f ,  f ,  and  f ,  to  equivalent  fractiofh  having  a 
common  denominator. 

3.  Beduce  f ,  f ,  f  to  equivalent  fractions  having  a  common 
denominator.  Ans.  ^i^,  \^l,  J-||. 

4.  Eeduce  ^\,  f ,  ^^  to  equivalent  fractions  having  a  com- 
mon denominator.  Ans.  4^a,  ^^%,  f  f^. 

5.  Eeduce  f ,  f,  |  to  equivalent  fractions  having  a  common 
denominator.  Ans.  ||8,  |o o,  |^. 

6.  Eeduce  -f^j  f ,  -^-^  to  equivalent  fractions  having  a  com- 
mon denominator. 

147.    Kule  for  reducing  fractions  to  equivalent  fractions  having  a  common 
denominator  ? 


114  COMMON    FRACTIONS. 

Though  the  above  rule  will  give  a  common  denominator, 
yet  it  will  not  always  give  the  least  common  denominator. 

Ex.  7.  Ecduce  |,  §,  ■^^,  i|  to  equivalent  fractions  having 
the  least  common  denominator. 

OPERATION. 

4)4,  8,  12,  16 

2 )  1,  2,      3,     4 

1,  1,      3,      2 

4  X  2  X  3  X  2  =r  48  =  L.  C.  M.  =  Least  Com.  Denom. 

(Art.  130,Eule.) 
48  -^  4  =  12,  and  12  X  3  zrr  3G  =  1st  numerator. 
48  -^  8  =  6,  and  6  X  5  =r  30  =  2d  numerator. 
48  -^  12  —  4,  and  4  X  7  =  28  r=  3d  numerator. 
48  -f-  16  —  3,  and  3  X  13  =  39  =  4th  numerator. 
Ans.,  If,  ff,  ff,  and  f|,  the  several  equivalent  fractions. 

Explanation.  We  first  find  the  least  common  multiple  of 
the  denominators  4,  8,  12,  and  16,  which  is  the  least  com- 
mon denominator.  We  then  divide  this  denominator  by  each 
of  the  denominators  of  the  given  fractions,  and  multiply  each 
quotient  b^  its  corresponding  numerator.      Hence  we  have  the 

KuLE.  Reduce  all  the  fractions  (if  necessary)  to  their 
lowest  terms.  Find  the  least  common  midtiple  of  all  the 
denominators  for  a  common  denominator.  Dicide  this 
multiple  hy  each  of  the  given  denominators,  and  multiply  the 
several  quotients  hy  their  respective  numerators  for  neio  num- 
erators. 

IToTE.  It  will  be  seen  that  both  the  above  rules  are  based  upon 
the  principle  Art.  139,  6th. 

147.  Rule  for  obtaining  the  least  common  denominator  of  several  fractions  ? 
On  What  principle  does  this  rule  and  the  former  one  depend  ?    Explain. 


COMMON    FRACTIONS.  115 

8.  Eeduce  f ,  t>  |»  i^  *^  equivalent  fractions  having  the 
least  common  denominator.  Ans.  ||,  §^,  f  |,  f  f . 

9.  Eeduce  f ,  ^,  |f  to  equivalent  fractions  having  the 
the  least  common  denominator. 

10.  Eeduce  f,  f,  ^^,  J  to  equivalent  fractions  having  the 
least  common  denominator. 

Case  9. 

14:8,  Numbers  that  are  of  the  same  kind  can  be  added 
together.  For  example,  6  pen?-|- 7  pens -(-5  pens  =:  18 
pens.  2  hats  -f-  5  hats  z=  7  hats ;  and  for  the  same  reason 
I  +  f  +  i  =r  f .  f  +  f ,  +  ^  ==  f ,  etc.  But  numbers  which 
are  not  alike,  cannot  he  added.  We  cannot  say  6  knives  -|-  4 
pens  =10  pens,  or  10  knives.  Neither  can  we  say  f  bush. 
-(-  f  qt.  =:  I  bush,  or  \  qt.  Numbers  must  be  of  the  same 
jiiND  if  we  would  add  them  together.     Hence, 

To  add  fractions  we  have  the  following 

Eule.  Reduce  the  fractions  to  equivalent  fractions  having 
a  common  denominator ;  after  which,  write  the  sum  of  the 
numerators  over  the  common  denominator. 

Ex.  1.     Add  -j^^,  y3^,  and  -J^  together.  Ans.  -^f. 

2.  Add  I,  |,  and  f  together.  Ans.  f§f  —  Ifgf. 

3.  Add  f ,  I,  ^\  and  f .  Ans.  f  f  f  =  2^^. 

4.  f  +  t  +  ^+|  =  what? 

5.  tV  +  l  +  f  +  i  =  what?     .    Ans.|H  =  2x||. 
^-     ^  +  §  +  1  +  1  =  what?  Ans.  5^2_i  ~  ^tV- 

7.  f  of  f -ft  of  i==^liat?  Ans.  If  =  1^1^. 
Note  1.     Compound  fractions  must  be  first  reduced  to  simple  frac- 
tions. 

8.  t  of  T^y  +  f  of  I  =  what  ?  Ans.  f |  =:  l^ 

9.  I  of  f  -(-i|of  f  =  what? 

148.    Can  numbers  not  alike  be  added  ?    Rule  for  adding  fractions  ? 


116  COMMON    FRACTIONS. 

10.  f  of  I  of  I  +  f  of  j\  =  what? 

11.  Add  41,  Gl,  and  of  together. 

4+6+5=:15,|  +  ^  +  fz:=2,y^,  aiidl5  + 
2TV^=rl7TV^,  Ans. 

Note  2.  Mixed  numbers  may  be  reduced  to  improper  fractions ; 
or  the  whole  numbers  and  the  fractions  may  be  added  separately, 
as  above,  and  tlien  their  sums  united. 

12.  Add  4i  and  7|  together.  Ans.  llf^. 

13.  Add  5f ,  7|,  and  f  together. 

14.  16|  +  14|  +  18f  z=  what  ? 

Practical  Questions  i:i  Addition  of  Fractions. 

1.  John  bought  a  top  for  i  of  a  dollar,  a  knife  for  |  of  a 
dollar,  and  a  "ball  for  i  of  a  dollar ;  how  much  did  they  cost 
him?  Ans.  $U. 

2.  Jane  bought  a  work-basket  for  |  of  a  dollar,  a  pair  of 
gloves  for  If  dollars,  and  gave  i  dollar  to  some  poor  persons  ; 
how  much  did  she  expend  in  all  ? 

3.  A  lady  bought  several  remnants  of  cloth.  One  piece 
was  I  yd.  long,  another  J  yd.,  and  a  third  ^  yd. ; 
how  much  cloth  did  she  buy  in  all  ?  Ans.  2^  yds. 

4.  A  coal  dealer  sold  coal  to  three  men,  as  follows :  To 
one  2|  tons,  to  another  5§  tons,  to  the  third  6^  tons;  how 
much  did  he  sell  to  all  ? 

6.  Three  men  bought  a  horse.  One  paid  51^  dollars, 
another  67f  dollars,  and  the  third  paid  76|-  dollars;  how 
much  did  the  horse  cost? 

Case  10. 

14:9.  The  remarks  under  Case  9  apply  with  equal  force  to 
questions  in  Subtraction.     We   cannot   take  4  pens  from  G 


COMMON    FRACTIONS.  117 

knives ;  nor  can  we  take  |  of  a  gallon  from  §  of  a  pound. 
Numbers  must  be  of  the  same  kind  or  the  subtraction  can- 
not be  performed.     Hence 

To  subtract  fractions  we  have  the  following 
EuLE.     Reduce  the  fractions  to  equivalent  fractions  having 
a  commcm  denominator^  and  tJien  write  the  difference  of  the 
numerators  over  tJie  common  denominator. 

Ex.  1.     From  I  take  |.  Ans.  f. 

2.  From  f  take  f.  Ans.  f. 

3.  From  f  take  f .  Ans.  /^. 

4.  From  f  take  y\.  Ans.  ff. 

5.  From  ^^  of  ^V  take  ^  of  f .  Ans.  ^Vf- 

6.  From  4|  take  2f .  Ans.  fj  z=  If  ^. 
Note  1.     Whenever  mixed  numbers  occur  in  the  question  they 

must  be  reduced  to  improper  fractions. 

7.  Subtract  f  from  Ih  Ans.  f f  —  Iff. 

8.  41  —  24=  what? 

9.  5y\  —  4f  =  what  ? 

10.  Ql-—6\=z  what  ?  Ans.  1  ^V- 

11.  I  of  iJ-— 2  of^^  — what? 

12.  f  of  5|  — f  of  S^zzzwhat? 

100.  Practical  Questions  in  Subtraction  of  Fractions. 
1.     A  boy  having  |  of  a  qt.  of  nuts,  gave  away  J  of  a  qt ; 
what  part  of  a  quart  had  he  left?  Ans.  ^|. 

2.  A  merchant  having  a  piece  of  cloth  containing  12f 
yds.  sold  7f  yds. ;  how  much  had  he  left  ?      Ans.  4f  f  yds. 

3.  A  farmer  had  a  field  containing  23 1  acres.  Of  this, 
6 1  acres  were  planted  with  corn,  and  the  remainder  bore 
grass ;  how  much  grass  land  was  there  in  the  field  ? 

Ans.  16ff  acres. 

149.    What  is  the  rule  for  subtraction  of  fractious  ? 


118  COMMON    FRACTIONS. 

4.  Bought  a  cask  of  wine  containing  37f  gal.;  16 1 
gallons  having  leaked  out,  what  quantity  remained  in  the 
cask  ?  Ans.  20^^!  gal. 

5.  A  boy  while  fisting  for  pickerel,  lost  part  of  his  pole  ; 
and  on.  measuring,  he  found  that  the  part  saved  was  |  of  the 
original  length.     What  part  was  broken  off  ? 

6.  From  a  chest  of  tea  weighing  62 1  lb.  39|  lb.  were 
sold.     How  many  pounds  remain  unsold  ? 

1^1.     Miscellaneous  Examples  in  Fractions. 

1.  Reduce  f  |  to  its  lowest  terms. 

2.  Ptcduce  7 1  to  an  improper  fraction. 

3.  Eeduce  -\^-  to  a  mixed  number. 

4.  Multiply  §^  by  9. 

5.  Multiply  jW  by  6. 

6.  Divide  |f  by  4. 

7.  Divide -V- by  10. 

8.  Divide  23^  by  7. 

9.  Multiply  20  by  f . 

10.  Multiply  100 1  by  I. 

11.  Multiply  f^  by  if. 

12.  Eeduce  f  of  f  of  -^^  to  a  simple  fraction. 

13.  Divide  ^9-  by  f . 

14.  Divide  207  by  |. 

15.  Eeduce  |-,   y^j,   -^^  to  equivalent  fractions  having  a 
common  denominator. 

16.  Addf  and  V-. 

17.  Add  ^^^V  and  i. 

18.  Add  10^  and  6f. 

19.  Addi,  §,|,  i 

20.  Subtract!  from  |. 

21.  Subtract  |  from  |§. 


COMMON    FRACTIONS.  119 

22.  Subtract  y\^  from  1. 

23.  Subtract  2i  from  3i 

24.  Keduce  |f,  -j^y^g*  1^1  *^  equivalent  fractions  with  a 
least  common  denominator. 

15.     Eeduce  -//j  of  ^V^  to  a  simple  fraction. 
Examples  in  Analysis. 

1^9.  We  analyze  an  example  when  we  solve  it  according 
to  its  own  conditions  without  being  guided  by  a^  particular 
rule. 

1.     If  1  pound  of  tea  cost  $1.20,  what  will  f  of  a  pound  cost? 

Analysis.  If  1  lb.  cost  $  2.20,  3^  of  a  pound  will  cost  ^  of 
$  1.20,  or  24  cts. ;  and  |^  of  a  pound  will  cost  4  times  24  cts., 
or  96  cts.,  which  is  the  answer. 

Note.  A  period  called  the  decimal  point,  which  will  be  here- 
after more  fully  explained,  is  placed  at  the  right  hand  of  dollars,  and 
the  first  two  places  at  the  right  of  the  points  always  express  cents. 
In  the  following  examples,  if  no  cents  are  named  with  the  dollars, 
their  places  may  be  supplied  with  two  ciphers. 

2.  If  1  cord  of  wood   cost  $  9,  what  will  f  of  a  cord 
cost?  Ans.  $5.62J. 

3.  What  will  f  of  a  ton  of  hay  cost  if  1  ton  cost  21 
dollars?  Ans.  $  15. 

4..    When  oil  is  $2.25   per  gall,  what  will  f  of  a  gall, 
cost?  Ans.  $1.50. 

5.  If  1    acre  of  land  cost  $  140,   what  will  -^j  of  an 
acre  cost  ?  Ans.  $  89.09 y^j. 

6.  If  I  of  a  yd.  of  cloth  cost  $  2.50,  what  will  be  the 
cost  of  1  yd.  ? 

Analysis.  If  f  of  a  yd.  cost  $  2.50,  }  will  cost  i  of 
$2.50,  or  $.83^  and  |  or  1  yd.  will  cost  5  times  $.83^, 
or  $  4.16f .,  which  is  the  answer. 

7.  Bought  f  of  an  iron  foundry  for  $6783,  what  was 
the  value  of  the  whole  foundry?  Ans.  $  8478.75. 


120  COMMOJ^    FRACTIONS. 

8.  "When  we  pay  8  62  for  ^  of  an  acre  of  land  what  is 
the  cost  per  acre  ?.  Ans.  $  70.85^. 

9.  If  I  of  a  gallon  of  molasses  cost  $.83,  what  is  the 
cost  per  gallon  ? 

10.  If  3 A.  2R.  30sq.  rd.  is  |  of  a  field,  what  is  the 
entire  area  ?  Ans.  6A.  2R.  22sq.  rd. 

11.  A  grocer  sold  from  a  cask  ISgal.  3qt.  Ipt.  of  oil, 
which  was^  of  what  the  cask  contained  ;  how  much  remained 
in  the  cask?  Ans.  21  gal.  Octt.  l^pt. 

Note.  It  is  evident  that  if  he  had  sold  3.  of  what  the  cask  at  first 
contained,  f  remained. 

12.  If  6  doz.  eggs  cost  $  1.68,  what  will  12|  doz.  cost? 

Analysis.  If  6  doz.  cost  $  1.68,  1  doz.  will  cost  ^  of 
$  1.68,  or  2Scts.  12|  doz.  =  \t  doz.  If  1  doz.  cost  28cts.  i 
doz.  will  cost  i  of  28cts.  or  7cts.,  and  -^^  will  cost  5 1  times 
7cts.,  or  $3.57.  Ans. 

13.  If  5  rods  of  wall  can  be  built  for  $  8.65  what  will  it 
cost  to  build  17|  rods  ?  Ans.  $  30.05|. 

14.  If  16  bushels  of  corn  cost  $22.72,  what  will  47§ 
bushels  cost  ?  Ans.  $  67.68f . 

15.  If  a  family  consume  4  barrels  of  flour  in  7  J  months, 
how  long  would  9  J  barrels  last  them  ?      Ans.  17^  months. 

16.  If  i  gal.  wine  cost  $  4.75,  what  will  6f  gal.  cost? 

Analysis.  If  J  gallons  of  wine  cost  $4.75,  |  will  cost  ^ 
of  $  4.75  or  $.67f ,  and  f  or  1  gallon  will  cost  8  times  67«  cts. 
or  $5.42f.  6 1  gallons  =  -\^  gallons.  Now  if  1  gallon  cost 
$5.42f,  -1  gallon  will  cost  ^  of  $  5.42f,  or  $  1.08f,  and  -\*- 
will  cost  34  times  $  1.08^  or  $  37.33f.  Ans. 

It.  When  f  of  a  dollar  will  purchase  3  qts.  of  cherries, 
how  many  can  you  purchase  for  2i  dollars  ?       Ans.  9|  qt. 


COMMON    FRACTIONS.  121 

18.  If  you  can  buy  4|  tons  of  hay  for  $  70,  what  will 
12|  tons  cost?  Ans.  $  188i 

19.  When  17^  cents  are  paid  for  If  lbs.  of  nuts,  how 
many  pounds  will  48|  cents  buy?  Ans.  4|  lbs. 

20.  If  6f  bushels  of  wheat  cost  $15,  how  many  bushels  can 
you  buy  for  68|  dollars  ? 

21.  Sold  5 1  cords  of  wood  to  one  man,  and  12f  cords  to 
another ;  how  much  did  I  sell  to  both  ? 

22.  How  many  hours  will  it  take  a  man  to  travel  65 f 
miles,  if  he  travel  3|  miles  in  an  hour  ?  Ans.  19^^  h. 

23.  Bought  a  horse  for  $176  J,  and  a  wagon  for  8  67| ; 
how  much  more  did  the  horse  cost  than  the  wagon  ? 

Ans.  $108j. 

24.  Paid  8  16  for  some  cloth,  at  the  rate  of  8  f  per  yd. ; 
how  many  yards  were  there  ?  Ans.  20  yards. 

25.  If  4  doz.  oranges  cost  f  of  $  4,  what  will  7  oranges 
cost?  Ans.  $  /o* 

26.  A  man  who  owned  |^  of  a  farm  sold  f  of  his  share  ; 
what  part  of  the  farm  did  he  sell  and  what  part  did  he  still 
own  ?  Ans.  sold  y^^,  and  had  left  ^|. 

27.  If  a  man  has  28f  gal.  of  wine,  and  sells  |  of  it, 
how  many  gallons  has  he  left?  Ans.  7f. 

28.  A  man  has  7  pieces  of  broadcloth,  each  piece  contain- 
ing 26|  yd.  This  he  makes  up  into  overcoats  requiring  4f 
yards  each  ;  bow  many  garments  can  he  make  and  how  much 
cloth  will  he  have  left  ?     Ans.  40  garments,and  {^yd.  left. 

29.  A  field  containing  157^sq.  rd.  is  9|rd.  wide;  how 
long  is  it?  Ans.  16|  rd. 

30.  If  f  of  a  ton  of  coal  can  be  bought  for  $  7,  what 
part  of  a  ton  can  be  bought  for  one  dollar  ?  Ans.  5^. 

31.  If  |-  of  a  ton  of  hay  is  worth  $  12|,  what  is  the 
value  of  a  ton?  Ans.  8  15|. 


122  COMMON    FRACTIONS. 

32.  One  man  earns  $  If  per  day,  and  another  earns 
$  2|- ;  how  much  will  they  both  earn  in  5  days  ? 

Ans.  $19^. 

33.  A  merchant  buys  flour  at  $  9f  per  barrel  and  sells 
it  for  $  12| ;  how  much  will  he  make  on  5  barrels? 

Ana.  e  15|. 

34.  A  tailor  made  3  suits  of  clothes,  each  containing  3f 
yards  of  cloth,  from  a  piece  35  yards  long;  how  much  was 
left?  Ans.  24fyd. 

35      What  will  12|  cords  of  wood  cost  at  $  8f  p3r  cord  ? 

Ans.  $109^^. 

36.  A  farm  containing  247  acres  was  sold  for  $  21118J  ; 
what  was  the  price  per  acre  ?  Ans.  $  85^^. 

37.  Four  partners  purchase,  goods  to  the  amount  of 
$  1264f,  and  sell  the  same  for  $  1586f.  The  profits  being 
divided  equally,  what  was  each  one's  share  ? 

Ans.  $  80xVu- 

38.  A  tailor  paid  $  12|  for  cloth,  and  $  5|  for  making  up 
the  same  into  a  coat,  vest,  and  pants,  and  sold  the  same  for 
$  26i ;  what  were  his  profits?  Ans.  $  7j\>-. 

39.  A  merchant  sold  to  a  customer  5|yd.  broadcloth, 
6|yd.  doeskin,  and  4|yd.  cassimere ;  how  many  yards  in  all 
were  there?  Ans.  16:|f. 

40.  When  flour  is  $  15  per  bbl.  how  many  barrels  can 
be  bought  for  46^  bushels  of  wheat  at  $  3^  per  bushel? 

Ans.  9|^  bbl. 

41.  A  person  owning  ^  of  a  ship,  sold  y\  of  his  share  for 
$  6000,  which  was  $  950  more  than  it  cost  him ;  what  did  he 
pay  for  his  entire  share  ? 

42.  A  gentleman  has  $  9750  invested  in  United  States 
Bonds,  which  is  f  of  his  fortune  ;  how  much  is  he  worth  ? 


DECIMAL  FRACTIONS.  123 


DECIMAL    FKACTIONS. 

1^3.  A  Decimal  Fraction  is  a  fraction  whose  denomi- 
nator is  10,  100,  1000,  or  1  with  one  or  more  ciphers  annexed. 

104.  The  denominator  of  a  Common  Fraction  may  be 
any  number  whatever.  Every  principle  and  every  operation  in 
Common  Fractions  is  equally  applicable  to  Decimals. 

\55,  The  denominator  of  a  decimal  fraction  is  not  usually 
expressed,  since  it  can  be  easily  determined,  it  being  1  with  as 
many  ciphers  annexed  as  there  are  figures  in  the  given  decimal. 

1^6.  A  decimal  fraction  is  distinguished  from  a  whole 
number  by  a  period,  called  the  decimal  point  or  scparatrix, 
placed  before  the  decimal ;  the  first  figure  at  the  right  of  the 
point  is  tenths  ;  the  second,  hundredths ;  the  third,  thousandths  ; 
etc. ;  thus,  .6  =:  -^^,  ,06  =  y§^,  .006  :r=  i^^is^  etc.,  the  figures 
in  the  decimal  decreasing  in  value  from  left  to  right,  as  in 
whole  numbers. 

\SK •  Since  whole  numbers  and  decimal  fractions  both 
decrease  by  the  same  law  from  left  to  right,  they  may  be  ex- 
pressed together  in  the  same  example,  and  numerated  as  in  the 
following 

NUMERATION  TABLE. 


(-1  cJ 

.S  -         ^  O 

TO-        .  CL  rS       ^        2       ^        °Q 


;2rt3  ..TO.SrQ-TSSE-lTj 


KhPRhWhh 


jT-'S 


5       9        8.       7       2 


vv: 


124  DECIMAL   FRACTIONS. 

158.  A  whole  number  and  decimal  fraction  written  to- 
gether, as  in  the  above  table,  form  a  mixed  number.  The 
integral  part  is  numerated  from  the  decimal  point  towards  the 
left,  and  the  fraction  from  the  same  point  towards  the  right, 
each  figure,  both  in  the  whole  number  and  decimal,  taking  its 
name  and  value  from  its  distance  from  the  decimal  point. 
Hence, 

159.  Moving  the  decimal  point  one  place  towards  the 
right,  multiplies  the  number  by  10;  moving  the  point  two 
places  multiplies  the  number  by  100,  etc.  Also  moving  the 
point  one  place  to  the  left,  divides  the  number  by  10  ;  moving 
the  point  two  places  divides  by  100,  etc. 

100.  A  decimal  is  read  like  a  whole  number,  giving  in 
addition,  the  name  of  the  denomination  of  the  right-hand 
figure  to  the  entire  number.  Thus,  .46  is  read  forty-six  hun- 
dredths ;  .073  is  read  seventy-three  thousandths;  .0068  is 
read  sixty-eight  ten  thousandths;  42.045  is  read  forty-two, 
and  forty-five  thousandths,  etc. 

101.  Since  multiplying  both  terms  of  a  fraction  by  the 
same  number  does  not  alter  its  value  (Art.  139,  5th),  annex- 
ing one  or  more  ciphers  to  a  decimal  does  not  affect  its  value; 
thus,  -fjs  —  j%o^  —  f^OoV  etc. ;   i.  e.  .2  =  .20  =  .200,  etc. 

lOS.  Prefixing  a  cipher  to  a  decimal,  i.  e.  inserting  a 
cipher  between  the  separatrix  and  a  decimal  figure,  diminishes 

153.  VThat  is  a  Decimal  Fraction  ?  154.  A  Common  Fraction,  what  is 
its  denominator  ?  Are  the  principles  of  common  fractions  applicable  to  deci- 
mals? 155.  la  the  denominator  of  a  decimal  usually  expressed?  15G.  How 
is  a  decimal  fraction  distinguished  from  a  whole  number  ?  What  is  the  first 
fi^re  at  the  right  of  the  point  ?  Second  ?  Third  ?  157.  Read  the  Numera- 
tion Table.  158.  What  is  a  mixed  number  ?  Which  way  is  the  integral  part 
numerated  ?  '  Which  way  the  decimal  ?  What  determines  the  name  and  value 
of  a  figure  ?  159.  How  does  moving  the  decimal  point  to  the  right  affect  the 
value  of  a  number?  How  moving  it  to  the  left 7  160.  How  is  a  decimal 
read?  Illustrate.  161.  How  does  annexing  one  or  more  ciphers  to  a  deci- 
mal affect  it  ?  Why  ?  16^.  Prefixing  one  or  more  ciphers  to  a  decimal  how 
affects  it?   Why? 


DECIMAL   FRACTIOITS.  125 

^e  value  of  that  figure  to  i^^j  its  previous  value ;  for  it  removes 
the  figure  one  place  further  from  the  decimal  point  (Art.  159)  ; 
thus,  .3  =z  ^jj  but  .03  =  only  y§^,  which  is  but  yV  of  yV 

Notation  and  Numeration  ov  Decimal  Fractions. 

103.  Let  the  pupil  write  in  figures  the  following  num- 
bers : 

1.  rifty-six  hundredths. 

2.  Eighty-seven  thousandths. 

3.  Two  hundred  sixteen  ten-thousandths. 

Note.  To  avoid  ambiguity  in  expressing  a  whole  number  and 
a  decimal,  we  should  use  "  and"  only  once,  and  that,  between  the 
whole  number  and  decimal,  as  two  hundred  three,  and  six  thou- 
sandths, (203.006)  ;  not,  two  hundred  and  three  and  six  thousandths. 
By  observing  the  above  rule,  much  trouble  will  bo  obviated,  and 
numerous  mistakes  avoided. 

4.  Twenty-eight,  and  one  thousand  nine,  ten- thousandths. 

5.  One  hundred  sixty-eight,  and  thirteen  millionths. 

6.  Eight    hundred    forty,  and   forty-two   hundred   thou- 
sandths. 

7.  Seven,  and  seven  hundredths. 

8.  Four  hundred  eighty-seven,  and  three  hundred  forty- 
four  ten-thousandths. 

9.  Sixteen  thousand,  four  hundred  thirty  nine,  and  ninety- 
two  thousandths. 

10.  Fifty-one  million  two  thousand  eighty-five,  and  seven- 
teen hundredths. 

11.  Four  thousand  eight  hundred  twenty-eight,  and  nine 
hundred  fifty-six  hundred-thousandths. 

12.  Eighty-seven  thousand  three  hundred  forty-nine,  ten- 
millionths. 

163.  How  are  operations  in  decimal  fractions  performed?  What  of  the 
the  decimal  point  ?    Proof? 


126  DECIMAL   FRACTIONS. 

KoTE  2.  Let  the  teacher  give  many  examples,  similar  to  the 
above,  continuing  the  exercise  till  the  pupil  can  write  decimals  with 
ease  and  accuracy. 

Numerate  and  read  the  following : 


1. 

.7 

11. 

4587.9506 

2. 

.03 

12. 

.000001 

3. 

40.6 

13. 

17.75851 

4. 

601.75 

14. 

7.805 

5. 

77.7 

15. 

84591.57 

6. 

9005.847 

16. 

42.2222 

7. 

.0001 

17. 

10000.001 

8. 

1000.1 

18. 

45671.3501 

9. 

45678.951 

19. 

1000.001 

0. 

.37558 

20. 

1846561.07 

As  all  the  operations  in  decimal  fractions  are  performed 
precisely  as  the  same  operations  in  whole  numbers,  no  explana- 
tions are  necessary,  except  to  determine  the  true  place  of  the 
decimal  point  in  the  several  results.  The  methods  of  proof, 
also,  are  the  same  as  in  whole  numbers. 

Case  1. 

164:.     To  add  decimal  fractions  ; 

Rule.  Place  tenths  under  tenths^  hundredths  under  hun- 
dredths^ etc.  ;  then  add  as  in  whole  numbers,  and  place  the  point 
in  the  sum  directly  under  the  points  in  the  numbers  added. 


Ex.  1. 

2. 

4.3  7 

4  6  9.037 

6  5.0  8  2 

609  3.0  08406 

9.0  9 

5  0  6.9  0  005 

4  6  3.0  8  0  4 

790  0.0  56209 

Sum, 

5  4  1.6  224 

Sum, 

149  6  9.00  1  6  65 

Proof, 

5  41.6  2  24 

Proof, 

1  4  9  6  9.0  0  1  6  6  5 

164.    Bole  for  addition  of  decimals  ? 


DECIMAL   FRACTIONS.  127 

3.  Add  469.05309,  27.039,  8056.00963. 

Ans.  8552.10172. 

4.  Add  904.0602,6095.8095,  600.06,  and  29076.004069. 

Ans.  36675.933769. 

5.  Add  2307.085063,  65.0047,  3S0.30027,  and  70580. 
060309.  Ans.  73332.450342, 

6.  Add  4.063,  85.605,  74608.37,  63.704. 

7.  Add  two  hundred  forty- three,  and  sixty-five  thousandths ; 
seventy-one,  and  eighty-four  ten-thousandths;  two  thousand, 
and  two  thousandths ;  six  hundred,  and  six  hundredths. 

Ans.  2914.1354. 

8.  Add  six  hundred  fifty-eight,  and  seven  hundred  two 
ten- thousandths ;  ninety- seven,  and  ninety-seven  hundredths; 
two  thousand  sixty-five,  and  eight  thousand  three  hundred  six 
hundred-thousandths. 

9.  Add  sixty-eight  millionths ;  two  hundred  forty-seven 
ten-thousandths ;  nine  hundred  seventy-two  hundred-thou- 
sandths. Ans.  .034488. 

10.  Add  three  thousand  nine  hundred  sixty- two,  and 'seven 
hundred  thousandths ;  five  hundred  seventy-three,  and  ninety- 
three  ten-thousandths;  eight  thousand  forty- four,  and  seven 
hundredths. 

Case  2. 
16^.     To  subtract  a  less  decimal  from  a  greater  : 

EuLE.  Place  the  less  number  under  the  greater,  tenths  under 
tenths,  etc.  ;  then  sithtract  as  in  whole  numbers,  and  place  the 
point  in  the  remainder ^  directly  under  the  points  in  the  minuend 
and  subtrahend. 

165.  Rule  for  Rubtraction  of  decimals?  When  the  number  of  decimal  places 
in  the  subtrahend  exceeds  the  number  of  decimal  places  in  the  minuend  what  is 
done? 


12^  DECIMAL   FRACTIONS. 


Ex.  1. 

2. 

3. 

5.9  2  3 

5  3.08  76 

4  9  6  3.0  0  3  2 

2.8  6  7 

47.19  84 

8  7  4.0  8  5  7  6  9 

Eem.    3.0  5  6 

5.8  8  92 

408  8.9  17  43  1 

Proof,  6.9  2  3 

6  3.087  6 

Note.  "Whenever  there  are  more  decimal  figures  in  the  subtra- 
hend than  in  the  minuend,  as  in  Ex.  3,  we  may  supply  the  deficiency 
by  annexing  ciphers,  or  supposing  them  annexed,  to  the  minuend. 

4.  From  68.0473  take  39.0027.  Ans.  29.0446. 

6.  From  234.0023  take  97.013005. 

G.  From  608.01004  take  290.020635. 

Ans.  317.989405. 

7.  From  5901.632  take  807.000956. 

Ans.  5094.631044. 

8.  From  20.006  take  7.020407. 

9.  From  one  hundred  eighty-three,  and  twenty-four  thou- 
sandths, take  seventy-six,  and  three  thousand  seven  hundred 
ninety-eight  ten-millionths.  Ans.  107.0236202. 

10.  From  five  thousand  six  hundred  nine,  and  one  hundred 
thirty-two  hundred-thousandths,  take  nine  hundred  eighty-five, 
and  four  hundred  ninety-six  ten-thousandths. 

Case  3. 

160.     To  multiply  one  decimal  by  another : 

EuLE.  Multi'ply  as  in  whole  numbers,  and  point  off  as  many 
figures  for  decimals  in  ike  product  as  there  are  decimal  places 
in  both  factors  counted  together. 

166.  Rule  for  multiplication  of  decimals  ?  Reason  of  the  rule  for  point- 
ing the  product  ?  Suppose  there  are  not  figures  enough  in  the  product  for 
observing  the  direction  for  pointing  off? 


DECIMAL   FKxVCTIONS.  129 

Ex.  1.  Multiply  .37  by  .28. 


OPEEATION. 

Multiplicand, 
Multiplier, 

.3.7 

.2  8 

296 
74 

PROOF. 


fo^TrXy^a^iVVoV 


Product,  .10  3  6 


Note.  1.  The  reason  of  the  rule  for  pointing  the  product  will  be 
obvious  if  we  change  the  decimals  to  the  form  of  common  fractions 
and  then  perform  the  multiplication ;  hence,  the  proof  above. 

Thus  we  have,  .24  X  .16  =  ^^^  X  t'o%  =  tM^  =  -0384. 
And     again,    .00G9  X  .09  =  y^6_9^^  X  toxt  =t<jUI'ipj  =^ 
.000621. 

.  2.  Multiply  3.0628  by  7.4. 

OPERATION.  PROOF. 

3.0  6  2  8  3.0628  =  Sj^-U<j  =  fSMf 
7  A  7.4        =7/^        =i# 


1   9  9  '^  1   9 

iiszioxz/  30628V'74 226  64  72 

2  14  3  9  6  T<rffo(J  ^  To — -lororifdo" 

2  2.6  64  72     2_2^66^7^4^2^22.foWD^o  ==22.66472.  Ans, 

3.  Multiply  .0638  by  .83.  Ans.  .052954. 

Note  2.  If  the  number  of  figures  in  the  product  is  less  than  the 
number  of  decimal  places  in  the  two  factors,  the  deficiency  must  be 
supplied  hy  prefixing  ciphers  to  the  product,  as  in  Ex.  3. 

4.  Multiply  56.029  by  .08506,  Ans.  4.76582674. 

5.  Multiply  289.406  by  56.09.  Ans.  16232.78254. 

6.  Multiply  368.09203  by  8.46. 

7.  Multiply  14.063  by  10.  Ans.  140.63. 

Note.  3.  By  referring  to  Art.  159,  it  will  be  seen  that  removing 
the  decimal  point  one  place  to  the  right  multiplies  the  number  by 
10,  etc. 


130  DECLAIAL   FRACTIONS. 

8.  Multiply  .0863  by  300. 

9.  Multiply  28.07  by  .03  Ans.  2.2456. 

10.  Multiply  4.7306  by  2.09. 

11.  Multiply  97.084  by  .063.  *  Ans.  6.116292. 

12.  Multiply  .75  by  .0024.  Ans.  .0018. 

13.  Multiply  803.006  by  .0001. 

14.  Multiply  .0G053  by  .0057.  Ans.  .000345021: 

15.  Multiply  119.79325  by  .006.  Ans.  .7187595. 

16.  Multiply  68.003  by  8.04. 

17.  Multiply  8.59  by  240.  Ans.  2061.6. 

18.  Multiply  .06  by  0003. 

19.  Multiply  .863  by  1000.  Ans.  863. 

20.  Multiply  3800  by  .046.  Ans.  174.8. 

21.  Multiply  6000  by  .006. 

22.  Multiply  .07  by  .07,  also  .5  by  .5. 

23.  If  the  multiplicand  is  642.  08069,  and  the  multiplier  is 
46.003,  what  is  the  product? 

24.  Tf  a  man  can  earn  S  64.925  in  1  month,  how  much  can 
he  earn  in  8.4  months?  '  Ans.  $  545.3*7. 

25.  If  1  barrel  of  potatoes  weighs  124.8  lb.,  how  much 
will  28.5  barrels  weigh?  Ans.  3556.8  lb. 

26.  If  a  trader  gains  $  .0625  on  one  pound  of  tea,  what 
will  he  gain  on  3000  pounds? 

27.  Should  the  same  trader  lose  $.875  on  1  bbl.  flour,  what 
would  be  his  loss  on  500  barrels?  Ans.  $  437.50. 

28.  If  it  require  6.75  yards  of  cloth  to  make  a  uniform  for 
a  soldier,  how  many  yards  will  it  take  to  furnish  4  regiments, 
of  1000  men  each? 

29.  If  a  horse  will  travel  46.875  miles  in  one  day,  how  far 
will  he  travel  in  four  weeks,  resting  on  each  Sabbath  ? 

Ans.  1125  miles. 

30.  When  $16.5  are  paid  for  1  ton  of  hay,  what  will  6.75 
tons  cost  ?  Ans.  $111,375. 


DECIMAL   FRACTIONS.  131 

Case  4. 
167.     To  divide  one  decimal  by  another : 
EuLE.     Divide  as  in  whole  numbers^  and  point  off  as  many 
Jigures  for  decimals  in  the  quotient  as  the  number  of  decimal 
places  in  the. dividend  exceeds  those  in  the  divisor. 
Ex.  1.  Divide  5.12  by  .8. 

OPERATION.  PROOF. 

.8  )  5.1  2  (  6.4  The  mixed  number  5.12  =  \^%, 

48  and,.8:=yV 

3  2  64 

3  2  512      8  _$a:^     £^_6^_ 

lOO~]'o'~I00^T~rO~     '^' 
10 

Note  1.  The  rule  for  determining  the  place  of  the  point  in  the 
quotient  may  be  explained  by  changing  the  decimals  to  the  form  of 
common  fractions  and  performing  the  division. 

2.  Divide  .000048  by  .03.  Ans.  .0016. 

Note  2.  If  the  number  of  figures  in  the  quotient  is  less  than  the 
excess  of  decimal  places  in  the  dividend  over  those  of  the  divisor, 
supply  the  deficiency  by  prefixing  ciphers  to  the  quotient. 

3.  Divide  420.075  by  25.  Ans.  16.803. 

4.  Divide  .19872  by  .276.  Ans.  .72. 

5.  Divide  34.944  by  .96. 

6.  Divide  36.75  by  .25. 

7.  Divide  .04212  by  4.68.  Ans.  .009. 

8.  Divide  1167.25  by  287.5.  Ans.  4.06. 

9.  Divide  44.8514  by  7.03.  Ans.  6.38. 

167.  Rule  for  division  of  decimals  ?  How  may  the  method  of  determining 
the  place  of  the  decimal  point  in  the  quotient  be  explained  ?  If  the  number  of 
fibres  in  the  quotient  is  less  than  the  excess  of  the  decimals  in  the  dividend: 
over  those  in  the  divisor,  what  is  to  be  done  ?  How  may  the  decimal  places 
of  dividend  and  divisor  be  made  equal  ?  What  is  then  the  quotient  ?  How 
is  a  remainder  sometimes  indicated  ? 


132  DECIMAL   FEACTIONS. 

10.  Divide  .06  by  .0002.  Ans.  300. 

11.  Divide  .5  by  .125.  Ans.  4. 

Note  1.  If  there  are  more  decimal  places  in  the  divisor  than  in 
the  dividend,  the  number  may  be  made  equal  by  annexing  one  or 
more  ciphers  to  the  dividend.  The  quotient  will  then  be  a  whole 
number. 

12.  Divide  43.6  by  7.2.  Ans.  6.+. 
Note  2.    When  a  remainder  occurs  in  the  division,  we  sometimes 

write  the  sign  -f-  after  the  quotient,  to  show  that  the  decimal  is  not 
complete,  or  we  may  annex  the  remainder  in  the  form  of  a  common 
fraction. 

13.  Divide  .875  by  .42.  Ans.  2.0|. 

14.  Divide  46.71  by  2.3.  Ans.  20.3  +. 

15.  Divide  58.996  by  4.5.  Ans.  13.11-f-. 

16.  Divide  .875  by  .875. 

17.  Divide  4.  by.  16. 

18.  Divide  .16  by  4. 

19.  28.75-^2.5.  Ans.  11.5.     " 

20.  436.8  ^  .74.  Ans.  590  +. 

21.  .08625  -i-  .005.  Ans.  17.25. 

22.  If  400  barrels  of  flour  cost  $  6700,  what  will  one  barrel 
cost?  Ans.  $16.75. 

23.  A  grocer  sold  a  quantity  of  tea ;  he  gained  $.125  on  a 
pound,  and  bis  whole  gain  was  $  3.75,  how  many  pounds  did 
he  sell?  Ans.  30. 

24.  If  a  chest  of  tea  holds  63.75  lb.,  how  many  chests  will 
it  require  to  hold  1912.5  lb.?  Ans.  30. 

25.  If  a  steamer  runs  617.5  miles  in  32.5  hours,  how  far 
does  she  go  in  1  hour  ?  Ans.  1 9  miles. 

26.  If  a  family  consume  6.5  barrels  of  flour  in  one  year,  how 
long  would  317.85  barrels  last  the  same  family? 

27.  A  man  paid  $  262.30  for  61  sheep,  how  much  did  he 
pay  apiece  ?  Ans.  $  4.30. 


DECIMAL   FRACTIONS.  133 

Case  5. 

168.     To  reduce  a  common  fraction  to  a  decimal : 

Ex.  1.  Keduce  |  to  a  decimal  fraction. 

If  we  multiply  a  fraction  by  any  number,  and  then  divide 
by  the  multiplier,  the  quotient  will  be  the  multiplicand.  Ac- 
cordingly, in  the  above  example,  we  multiply  |  by  1000  = 
ILOJLQ.—  Q26  ;  and  625 -i- by  1000::=  ^%y>Q-=z.(j25  ;  and  hence 
we  have  the  following 

EuLE.  Annex  one  or  more  ciphers  to  the  numerator  and  di- 
vide the  result  by  the  denominator,  continuing  the  operation 
until  there  is  no  remainder,  or  as  far  as  is  desirable.  Point  off 
as  many  decimal  places  in  the  quotient  as  there  are  ciphers 
annexed  to  the  numerator, 

2.  Eeduce  |  to  a  decimal. 

I  X  100  -=  iao  _  -J5  .  and  7^  -^  100  =  .75  Ans. 

3.  Reduce  f^  to  a  decimal.  Ans.  .5625. 

4.  Reduce  |^f  to  a  decimal.  Ans.  1.171875. 

5.  Reduce  f  |  to  a  decimal. 

6.  Reduce  -^^  to  a  decimal.  Ans.  .4166  -f-. 

7.  Reduce  ^,  f,  f,  f,  |,  ^^  to  decimals. 

Case  6. 
109.     To  reduce  a  decimal  to  a  common  fraction  : 
Ex.  1.  Reduce  .75  to  a  common  fraction.     .75=:^^^^,  and 
this,  reduced  to  its  lowest  terms  z=  |  Ans.     Hence, 

Rule.  Write  the  denominator  to  the  decimal,  omitting  the 
decimal  point,  and  then  reduce  the  common  fraction  to  its  lowest 
terms, 

2.  Change  .625  to  a  common  fraction.  Ans.  f . 

168.    Rule  for  reducing  a  common  fraction  to  a  decimal  ?     169.    Rule  for 
reducing  a  decimal  to  a  common  fraction  ? 


134  DECIMAL,   ru ACTIONS. 

3.  Change  .375  to  a  common  fraction  and  to  its  lowest 
terms.  Ans.  |. 

4.  Keduce  .0625  to  a  common  fraction. 

Ans.  ^^^. 

5.  What  common  fraction  is  equivalent  to  .4375? 

Ans.  ^^. 

6.  Eeduce  .68  to  a  common  fraction. 

7.  Change  .875  to  a  common  fraction. 

8.  Change  .0075  to  a  common  fraction. 

Miscellaneous  Examples   in   Decimals. 

1.  What  is  the  sum  of  one- tenth,  one  hundredth,  and  for- 
ty-seven thousandths?  Ans.  .157. 

2.  What  is  the  difference  between  seven  hundredths  and 
eight  thousandths?  Ans.  .062. 

3.  .065  — .0098  z=  what? 

4.  Multiply   eighty-four  hundredths   by   forty-seven  ten- 
thousandths.  Ans.  .003948. 

5.  Divide  two  by  four-tenths. 

6.  Divide  eighteen  thousandths  by  six  millionths. 

Ans.  3000. 

7.  From  seven- tenths  take  four  millionths. 

Ans.  .699996. 

8.  Paid  $  480  for  a  piece  of  land  at  $  62.50  per  acre ;  how 
many  acres  were  there  ?  Ans.  7.68. 

9.  What  cost  8  acres  of  land  at  $  68.75  per  acre? 

10.  What  cost  6.75  lb.  of  coffee  at  $  .24  per  lb.  ? 

11.  How  many  casks  each  holding  37.5  gallons  can  be  filled 
"with  168'?. 5  gallons  of  wine? 

12.  Bought  6.5  tons  hay  at  $  16.875  per  ton;  what  was 
the  entire  cost?  Ans.  $  109.68.75. 

13.  What  common  fraction  is  equal  to  the  sum  of  .625  and 
.0625  ?  Ans.  ^. 


DECIMAL   FRACTIONS.  135 

14.  When  $  18.5625  is  paid  for  148.5  yds.  of  cloth,  what  is 
the  cost  per  yard  ? 

15.  What  will  17  pairs  of  boots  cost  at  $  10.875  per  pair? 

16.  Change  .68  to  a  common  fraction.      Ans.  ^^^  ^^il- 

17.  Change  5.25  to  a  common  fraction. 

Ans.  if^^-V-^Si- 
18.  How  many  pairs  of  shoes  at  $1.25  can  be  purchased 

for  $  45  ? 

1^.  How  many  pounds  of  sugar  at  8  .18  per  pound  can  you 

buy  for  $6.12?  Ans.  34. 

20.  What  will  12  bales  of  cotton  cost,  each  bale  weighing 
5.25  cwt.  at  $  46.50  per  cwt.? 

21.  If  .625  of  a  ton  of  coal  cost  $5. 75,  what  will  one  ton 
cost?  Ans.  $9.20. 

22.  What  cost  .875  of  a  ton  of  coal  at  8  12  per  ton  ? 

xVns.  $  10.50. 

23.  What  will  8.75  cords  of  wood  cost  at  $  10  per  cord? 

24.  If  a  boat  will  sail  7.5  miles  in  1  hour,  how  far  will  she 
sail  in  9.75  hours? 

25.  Divide  one  hundred  by  one  hundredth. 

2Q.     Multiply  one  thousandth  by  one  thousandth. 

27.  If  9564.75  rods  of  wall  can  be  built  in  87.75  days, 
how  many  rods  can  be  built  in  one  day  ?  Ans.  109. 

28.  I  have  a  room  15.50  feet  wide,  16.75  feet  long;  how 
many  square  feet  does  the  floor  contain?        Ans.  259.625. 

29.  How  many  square  yards  of  carpeting  would  it  take  to 
carpet  the  above  room  ? 

30.  How  much  would  the  above  carpet  cost  at  $  1.75  per 
yard  ? 

31.  A  load  of  hay  weighs  1675.25  lb.  how  much  will 
it  cost  at  8  2.50  per  cwt.?  Ans.  $  41.88. 

32.  A  ship  carries  725  bales  of  cotton,  each  bale  weighing 
400  pounds;  how  much  will  the  freight  amount  to  at  $.0125 
per  pound  ? 


136  UliTITED   STATES   MONEY. 


UNITED    STATES    MONEY. 

ITO.  United  States  Money,  sometimes  called  Federal 
Money t  is  the  currency  of  the  United  States. 

TABLE. 

10  Mills  (m.)  make  1  Cent,  Marked  c. 

10  Cents                "  1  Dime,                  ''  d. 

10  Dimes              "  1  Dollar,                "  % 

10  Dollars             "  1  Eagle,                ''  e. 

Cents.  Mills. 

Dimes.                   1  =  10 

Dollars.              1     —         10  =  100 

Eagle.            I     —        \0     =        100  =  1000 

1     :=     10     =     100     =     1000  =  10000 

Note.  The  terms  eagle  and  dime  are  seldom  or  never  used  in  com- 
putation ;  eagles  and  dollars  being  read  collectively  and  called  dollars, 
and  dimes  and  cents  being  called  cents  ;  thus,  3  eagles  and  5  dollars 
are  called  $  35,  and  4  dimes  and  3  cents  are  called  43  cents.  When 
mills  are  written  with  dollars  and  cents  they  are  set  in  the  third  place 
at  the  right  of  the  period;  thus,  thirty-five  dollars,  forty-three  cents, 
and  seven  mills,  expressed  in  figures,  is  ^  35.437. 

171.  A  coin  is  a  piece  of  gold,  silver,  or  other  metal, 
stamped  by  authority  of  the  General  Government,  to  be  used 
as  money. 

173.  The  coins  authorized  by  our  Government,  and 
stamped  at  the  U.  S.  Mint,  are  the  following: 

170.  What  is  United  States  Money  ?  Repeat  the  Table.  Are  the  terms 
eagle  and  dime  much  used  ?  How  are  eagles  and  dollars  read  ?  Dimes  and 
cents  ?    What  place  do  mills  occupy  ?    Illustrate.    171.    What  is  a  coin  ? 


UNITED   STATES   MONEY.  137 


Gold 

Silver. 

Double  Eagle. 

S  20.00 

Dollar, 

$1.00 

Eagle, 

10.00 

Half  Dollar, 

.50 

Half  Eagle, 

5.00 

Quarter  Dollar, 

.25 

Quarter  Eagle, 

2.50 

Dime, 

.10 

Three  Dollar  Piece. 

3.00 

Half  Dime, 

.05 

One  Dollar, 

1.00 

Three  Cent  Piece, 

.03 

Also  of  copper,  bronze,  and  nickel,  we  have  the  One  Cent, 
Two  Cent,  Three  Cent,  and  Five  Cent  Pieces. 

Note  1.     The  mill  is  not  coined. 

Note  2.  Our  Government,  by  enactment  of  Congress,  may  recall 
any  of  these  coins,  or  issue  new  ones  of  different  values  and  com- 
posed of  different  metals,  at  any  time. 

Note  3.  The  greater  part  of  the  currency  in  general  use  in  this 
country,  consists  of  bank  hills  and  notes  of  the  General  Government^ 
which  are  much  more  convenient  for  most  purposes  than  gold  and 
silver. 

Operations  in  United  States  Money  are  performed  precisely 
like  those  in  Decimal  Fractions,  the  dollar  being  considered 
the  UNIT.     Therefore  no  special  rules  are  needed. 

Practical  Examples. 
1.  Paid  $  12.50  for  a  barrel  of  flour,  %  2.375  for  a  box  of 
sugar,  $  17.875  for  a  tub  of  butter,  and  $  5.25  for  a  cheese  ; 
what  did  I  pay  for  all  ?  Ans.  $  38. 

Having   set   dollars  under  dollars, 

o  o  7  r  cents  under  cents,  etc.,  add  as  in  Art. 

-      '^  „  ^  164,  and  set  the  sum  below,  remember- 

5  2  5  i'^o  ^^^^  ^^®  point,  or  period,  should 

TT—— — — -  be  placed  directly  under  the  points  in 

$3  8,0  0  0,  Ans.    ^,    ^       ,  ,/,  ^ 

the  numbers  added. 

17ii.  What  gold  coins  are  authorized  by  our  Government  ?  What  silver 
coins  ?  What  other  coins  ?  What  is  said  of  the  mill  ?  What  of  changing  the 
coins  in  use  ?  What  of  paper  money  ?  How  are  operations  in  U.  S,  Money 
performed  ? 


138  UNITED   STATES   MONEY. 

2.  Bought  a  coat  for  $  21.75,  a  vest  for  $  5.35,  a  pair  of 
pantaloons  for  S  8.40,  a  hat  for  $  5.25,  a  pair  of  boots  for  $  7.50, 
and  various  other  articles  for  $12.75  ;  what  must  I  pay  for  all?- 

3.  A  farmer  paid  $  125.50  for  a  pair  of  oxen,  $  52  for  a 
cow,  $350.75  for  a  horse,  and  $45.25  for  a  harness;  how 
much  did  all  costf  Ans.  $  573.50. 

4.  A  merchant  in  returning  from  the  city  found  he  had 
expended  $  1050.375  for  dry  goods,  $  850.75  for  groceries  and 
$  250.875  for  hardware ;  what  was  the  amount  of  his  purchases  ? 

5.  A  broker  has  $  19505.00  in  one  bank,  $4550.50  in 
another  bank,  and  $  6750.37  in  another  ;  how  much  money  has 
he  in  the  three  banks  ?  ,  Ans.  $  30805.87. 

6.  The  property  of  a  gentleman  is  divided  as  follows ;  he 
has  $5750  in  bank  stock,  $3100.50  in  notes  at  interest,  a 
manufactory  worth  $  19587,  two  farms,  one  worth  $  5780  and 
the  other  twice  as  much,  and  $  6850.75  due  him  on  accounts; 
how  much  is  he  worth  ? 

7.  A  man  who  owed  $87.37,  paid  $16.52;  how  much 
did  he  still  owe  ? 

OPERATION.  Having    set   the    less   number    under    the 

8  7.3  7  greater,  dollars  under  dollars,  etc.,  subtract  as 

1  ^-5  2  in  Xvi.  165,  remembering  to  place  the  point  in 

$  7  0.8  5,  Ans.  the  remainder  under  the  points  in  the  minuend 
and  subtrahend. 

8.  Paid  $  175.625  for  a  pair  of  oxen,  and  $  132.375  for  a 
horse ;  how  much  more  did  I  pay  for  the  oxen  than  for  the 
horse?  Ans.  $43.25. 

9.  A  gentleman  purchased  a  city  residence  for  $  19570, 
which  was  $8957.75  more  than  his  country  place  cost  him; 
what  did  his  country  place  cost  him  ? 

10.  A  drover  paid  out  for  cattle  $  5767.50 ;  he  received 
for  the  same  lot,  besides  expenses  of  taking  them  to  market, 
$  6530  ;  how  much  were  his  profits?  Ans.  $  762.50. 


UNITED   STATES   MONEY.  139 

11.  A  merchant  went  to  the  city  to  buy  goods,  with 
$  3575.50  in  cash  ;  he  bought  to  the  amount  of  $  5050  ;  how 
much  did  he  buy  on  credit?  Ans. ^  1474.50. 

173.  To  find  the  cost  of  any  number  of  things 
when  the  price  of  one  thing  is  given. 

12.  Bought  6  cows  at  $35,375  each;  what  did  I  pay  for  the 
lot? 

OPERATION.  Six  cows  will  evidently  cost  6  times  as 

$  3  5.3  T  5  much  as  one  cow.  All  similar  examples  are 

p  solved  in  like  manner.      Hence,  the  fol- 

S  2  1  2.2  5  0,  Ans.    lowing 

EuLE.     Multiply  the  price  of  one  by  the  number. 

13.  "What  is  the  cost  of  7  barrels  of  flour  at  $8.50  per 
barrel  ? 

14.  Bought  IG  yards  of  silk,  at  $1.75  per  yard;  what  was 
the  cost  of  the  piece?  Ans.  $  23.00. 

15.  Bought  33  sheep,  at  $8.25  per  head ;  what  was  the  cost 
of  the  flock  ?  Ans.  $  272.25. 

16.  What  are  85  pounds  of  butter  worth,  at  3T  cents  per 
pound?  Ans.  $31.45. 

17.  "What  are  625  cords  of  wood  worth  at  $  8.75  per  cord? 

Ans.  $  5468.75 

18.  What  is  a  cargo  of  coal  of  2070  tons  worth  at  $  11.25 
per  ton?  Ans.  $23287.50. 

19.  What  are  1625  bushels  of  wheat  worth  at  $3.75  per 
bushel?  What  would  be  the  freight  on  the  above  at  $.625 
per  bushel  ? 

20.  Supposing  the  above  wheat  to  make  425  barrels  of 
flour,  how  much  would  it  be  worth  at  $  17.50  per  barrel? 

Ans.  $7437.50. 


173.    How  can  you  find  the  cost  of  any  number  of  things  when  you  know 
the  price  of  one? 


140  UNITED    STATES    MONEY. 

174L.     To  find  the  price  of  an  article  when  the  cost 
of  a  given  number  of  articles  is  known. 

21.  Paid  $1129.50  for  9  horses;  what  was  the  average  price 
per  horse  ? 

OPERATION.  One  horse   is  worth   one  ninth  as 

g\  2  129. 50  much  as  9  horses.     To  obtain  one- 

.  ninth  of  any  number  we  divide  the 

*  number  by  9.     Hence,  the 

Rule.     Divide  the  cost  hy  the  number. 

22.  Paid  $19.61  for  53  pounds  of  butter ;  what  was  the 
price  per  pound  ?  Ans.  ^7  cents. 

23.  If  I  pay  $315.75  for  25  barrels  of  flour,  what  is  the 
price  per  barrel  ?  Ans.  $  12.63. 

24.  If  a  mechanic  earns  $  65.24  in  28  days,  what  is  his  daily 
•wages? 

25.  Paid  $74.75  for  13  weeks'  board;  what  was  the  price 
per  week?  '  Ans.  $5.75. 

26.  Seventy-seven  boys  paid  $2233  for  1  year's  tuition; 
what  did  each  boy  pay  ?  Ans.  $  29. 

27.  Bought  a  farm  containing  87  acres  for  $4763.25  ;  what 
was  the  price  per  acre  ? 

28.  If  8  barrels  of  flour  cost  $75,  what  is  the  price  per 
barrel?  Ans.  $9.3T5. 

OPERATION.  When  the  division   is  incomplete 

8  )  $  7  5.0  0  0  and  there  are  no  cents  and  mills  in 

$  9.3  7  5,  Ans.        the  dividend,  ciphers  may  be  annexed 
to  the  dividend  and  the  division  continued. 


1T4.    How  can  you  find  the  price  of  one  article  when  you  know  the  cost  of 
a  given  number  of  articles  ?    Explain  Ex,  28. 


f  v^  <^ 


UNITED   STATES   MONEY.      ^  w  ST  *    niSfc  ^"^ 

\  Cj  ^      ^^ 

PROOF.  •        Ke versing  the  above  p^^t^MpFiO^ 

$  9.3  7  5  the  proof,  will  cause  the  ccnl 
8             mills  to  disappear  and  bring  back  the 


$  7  5.0  0  0  original  dividend. 

29.  If  21:  men  earn  $  63  in  a  day,  what  will  1  man  earn  in 
the  same  time?  Ans.  $2.G25. 

30.  Paid  S  6300  for  36  horses ;  what  was  the  price  of  each  ? 

31.  If  5  barrels  of  flour  are  worth  $47,267,  what  is  1  barrel 
worth?  Ans.  $9,453+. 

32.  Paid  $  34.88  for  9  yards  of  cloth ;  what  was  the  price 
per  yard?  Ans.  $3,875+. 

33.  If  a  cargo  of  wood  is  worth  $19275,  and  the  number 
of  cords  is  2850,  how  much  is  the  price  per  cord? 

Ans.  $  6.76+. 

34.  If  $  1000  will  buy  850  bushels  of  com,  what  is  the 
price  per  bushel  ?  Ans.  $1.17+. 

35.  If  a  merchant's  bill  for  flour  was  $18500  in  one 
month,  and  he  purchased  1500  barrels,  what  did  it  cost  him, 
on  an  average,  per  barrel? 

175,  To  find  the  quantity  when  the  cost  of  the 
quantity  and  the  price  of  one  are  given. 

36.  At  $8  a  ton,  how  many  tons  of  coal  can  I  buy  for 
$240? 

OPERATION. 

$8')S240  '^^  many  times  as  $  8  is  contained  in 

$  24  so  many  tons  I  can  buy.     Hence,  the 

3  0,  Ans. 

BuLE.     Divide  the  cost  hy  the  price  of  one. 


175.    How  do  you  find  the  quantity  when  the  total  cost  and  the  price  of 
one  are  given  ? 


142  UNITED    STATES   MONEY. 

37.  At  16  cents  a  pound,  how  many  pounds  of  sugar  can  I 
buy  for  $  19.96  ?  Ans.  124|. 

38.  How  many  yards  of  cloth,  at  $  2.56  per  yard,  can  I  buy 
for  $642.56?  Ans.  251. 

39.  How  many  sheep,  at  $  7.75  a  head,  can  be  bought  for 
$193.75?  Ans.  25. 

40.  A  farmer  paid  $  3562.50  for  land,  at  $  37.375  per  acre ; 
how  many  acres  did  he  buy  ? 

41.  A  merchant  paid  $  4498.83  for  a  lot  of  broadcloth ; 
the  average  price  per  yard  was  $  3.33  ;  how  many  yards  were 
there  ?  Ans.  1351. 

42.  How  many  books  at  $1.75  each,  can  be  bought  for 
$2625? 

43.  An  agent  has  $  925  with  which  to  purchase  flour  ;  at 
$  12.25  per  barrel  how  many  whole  barrels  can  he  buy  ?  How 
much  money  will  he  then  have  left? 

176.  To  find  the  cost  or  value  of  any  number  of 
articles  when  the  price  of  one  is  an  exact  or  aliquot 
part  of  a  dollar. 

Table  of  Aliquot  or  Exact  Parts  op  a  Dollar. 

50     cents  :=  ^  of  a  dollar,  20     cents  =  ^  of  a  dollar, 

331  cents  :=  ^  of  a  dollar,  16f  cents  =  ^  of  a  dollar, 

25     cents  =  :J-  of  a  dollar,  12^  cents  =  |  of  a  dollar. 

44.  What  cost  45  yards  of  calico,  at  33^  cents  per  yard  ? 
331  cents  is  ^  of  a  dollar ;  hence,  45  yards  will  cost 
$45-^3  =  $15,  Ans. 

45.  What  cost  84  pounds  of  butter,  at  50  cents  a  pound  ? 

46.  What  cost  48  pounds  of  honey,  at  25  cents  a  pound  ? 

47.  What  cost  32  bushels  of  corn,  at  87^  cents  per 
bushel? 


176.    What  is  an  aliquot  part  ?    Repeat  the  table  of  aliquot  or  exact  parts 
of  a  dollar. 


UNITED   STATES   MONEY.  143 

OPERATION. 

$32  =  cost  of  32  bush.,  at  S  1. 


16=  cost  of  32  bush.,  at  50  c,  ot  J-  of  $  1. 
8  :=cost  of  32  bush.,  at  2  5  c,  or  ^  of  50c. 
4  =  cost  of  32  bush.,  at     12^  c,  or  ^  of  25c. 

Ans.  f2S  =  cost  of  32  bush.,  at     sTi  c. 

That  is,  the  cost  at  $  1  is  evidently  as  many  dollars  as  there 
are  bushels ;  the  cost  at  50c.,  is  half  as  much  as  at  $  1 ;  the 
cost  at  25c.,  half  as  much  as  at  50c. ;  and  the  cost  at  12^c., 
half  as  much  as  at  25c.  Then  the  cost  at  50c.,  at  25c.,  and 
at  12Jc.,  added,  gives  the  cost  at  87ic. 

48.  What  is  the  value  of  736  yards  of  gingham,  at  37^  cents 
a  yard  ?  Ans.  $  276. 

49.  What  shall  I  pay  for  1832  bushels  of  oats,  at  62i  cents 
per  bushel  ? 

This  process  is  usually  called  Praciice,{oT  which  we  have  the 
following 

KuLE.  Take  such  cdiQUot  pqrts  of  the  number  of  articles  as 
the pHce  is  an  aliquot  part  of  %\, 

50.  What  cost  24  barrels  of  apples,  at  %  3.75  per  barrel? 

OPERATION. 

$2  4  =  cost  at  S  1. 
_3 

$7  2  =  costat$3. 
12  =  cost  at      .5  0  or  J-  of  $  1. 
6  =  cost  at       .2  5  or  I  of  50c. 

Ans.  $90  =  cost  at  $  3.7  5. 

51.  What  are  348  barrels  of  flour  worth,  at  %  9.87^-  per 
barrel?  Ans.  $3436.50. 

52.  What  are  165  thousand  of  brick  worth,  at  $  11.75  per 
thousand  ? 


176.    How  do  you  find  the  cost  of  any  number  of  articles  when  the  price  is 
an  aliquot  part  of  a  dollar  ?    What  is  this  process  called  ? 


144  UNITED   STATES   MONEY. 

53.  Wliat  are  84  cases  of  mercliandise   worth,    reckoning 
each  case  at  $  475.3 7^? 

54.  What  would  be  the  cost  of  336  yards  of  carpeting  at 
$2.G2J  per  yard?  Ans.  $  882. 

65.  How  much  would  1250  cords  of  wood  cost  at  $  8.75 
per  cord  ?  Ans.  8  10937.50. 

177,     To  exchange  or  barter  goods. 

6Q.  How  many  pounds  of  sugar,  at  20  cents  a  pound,  shall 
I  give  for  50  bushels  of  com,  at  80  cents  a  bushel  ? 

OPERATION. 

4  This  example  is  best  solved 

$  0  X  5  0  by  cancelling  as  in  the  margin. 

(\1\        ^^  ^^^'  "^^^  ■''■*'  ^^^  ^^^^  ^^  analyzed  as  fol- 

^  ^  lows  ;    50  bushels  at   80   cents 

are  worth  50  times  80  c.  =  4000g.,  and  20c.  in  4000c.,  200 
times,  the  number  of  pounds  of  sugar  required,  Ans.  200. 

57.  How  many  cords  of  wood,  at  $  8  per  cord,  will  pay  for 
6  tons  of  hay,  at  $20  per  ton?  Ans.  15. 

58.  How  many  tons  of  coal,  at  %  12.50  per  ton,  will  pay  for 
16  yards  of  cloth,  at  %  6.25  por  yard?  Ans.  8.  • 

59.  How  much  flour  at  $  10.50  per  barrel  can  be  obtained 
for  150  bushels  of  potatoes  at  75c.  per  bushel  ? 

60.  How  many  entire  yards  of  broadcloth  at  $  5.75  per 
yard,  can  be  bought  for  3^  cords  of  wood  at  $  6.75  per  cord, 
and  what  money  will  remain  duo  ?    Ans.  4  yards,  and  62^c.  due. 

61.  How  many  bushels  of  wheat  at  $  3.50  per  bushel  would 
purchase  50  bushels  of  corn  at  $1.50. per  bushel?      Ans.  2\f. 

62.  It  requires  16  thousand  shingles  to  cover  the  roof  of  a 
certain  house,  and  they  cost  $4.75  per  thousand;  how  many 
days  work  at  %  2.50  per  day  would  it  require  to  pay  for  them  ? 

177.    Explain  the  operation  in  Ex.  56  by  cancellation.    How  else  mfey  this 
example  be  solved  ? 


UNITED   STATES   MONEY.  145 

63.  How  many  yards  of  cloth,  at  $  3.50  per  yard,  can  be 
had  for  3^  tons  of  hay,  at  $19.50  per  ton  ? 

BILLS. 

178.  A  Bill  of  Goods  is  a  written  statement  of  articles 
sold,  giving  the  price  of  each  article  and  the  cost  of  the 
whole. 

An  Account  is  a  written  statement  of  the  items  of  debt  and 
credit  between  two  persons  or  companies. 

The  person  or  company  who  owes  is  the  Debtor,  and  the 
one  to  whom  something  is  due  is  the  Creditor. 

When  a  bill  is  paid  it  is  usually  receipted  or  signed  by  the 
(Creditor  or  by  his  authorized  agent. 

Receipts  for  an  amount  of  $20  dollars  or  upwards,  according  to 
the  laws  of  Congress,  now  require  a  revenue  stamp  to  be  affixed. 

Find  the  cost  of  the  several  articles,  and  the  amount 
or  footing  of  each  of  the  following  bills. 


(1.) 

Mr.  John  Low, 

2  5  lb.  Sugar, 
4  2  lb.  Butter, 
1  5  yd.  Cloth, 

Boi 

Bought  of 
at 

n 

(( 

ved  Payment, 

tton,  Sept.  6,  186T. 

David  Flint, 
16  c. 
2  5  c. 
$3.3  3i 

1 
! 

Stamp. 

1 

Recei 

1 

$64.5  0 
David  Flint. 

178.  What  is  a  Bill  of  Goods  ?  What  an  Account  ?  Who  is  Debtor  ? 
Who  Creditor?  When  should  a  bill  be  signed  or  receipted?  By  whom? 
What  is  said  of  affixing  a  revenue  stamp  ? 


146                    unit:ed  states  money. 

(2.)  New  York,  Oct.  15,  1867. 
Messrs.  Smith  &  Co., 

Bought  of  Abel  Adams, 

2  4  gal.  Molasses^  at            8  7 1^  c. 

3  2  gal.  Syrup,  "        $  1.1 2 1 

4  8/5.  Coffee,  "              ^7^c. 
16  lb.  Tea,  **             6  2ic. 


Stamp. 


8  8  5, 
Received  Payment, 

Abel  Adams, 

By  L.  Snow. 


(3.)  iVew'  Orleans,  Dec.  19,  1865. 

3/r.  James  Fitoh, 

1865.  To  Henry  Day  &  Co.,  Dr. 

June    4.     To  12  Day's  Algebras,       at         S7^c. 
Aug.    9.      '*       4:  Beams  Paper,  "  $2.75 

16.      "    2  4.S'/a^es,  "         3  7ic. 

JVbv.  11.      *'        3  Webster's  Diction- 
aries, at  $  8.7  5 


Stamp, 


$5  6.7  5 
Received  Payment, 

John  Smith, 
For  Henry  Day  &  Co» 


(4.)  Norwich,  July  5,  1867. 

Mr.  R.  B.  Allen, 

Bought  of  James  Robinson  &  Co. 

1  2  pairs  Men's  Calf  Boots,     at  $  4.7  5. 

12     "         "       Thick  "  ♦*  3.7  5. 

18     "      Boys'     "       '*  "  2.1 2  J. 


Stamp.!  Received  Payment, 

I  James  Robinson  &  Co. 


UNITED   STATES    MONEY. 


147 


(5.) 
Messrs.  John  P.  Jones  &  Co., 


1866. 


Baltimore,  Dec,  15,  1866. 


To  E.  C.  Johnson  «&  Co.,  Dr. 


Mar.  4.  T'o  1  0  tons  Ice,  at 

Jpr.  8.  "25  hbl.  Flour, 

June  8.  "10  0  bush.  Com, 

"     "  ♦<    5  0  bush.  Wheat,  " 


1866.  Cr, 

May  14.  By  Cash, 

June    8.  "    ^Merchandise, 

Sept.    6.  "     5  cords  Oak  Wood,     at 

Dec,     1.  "     2  tons  May,  ** 


Stamp. 


$12.37^ 
9.2  5 
8  7c. 
81.75 


$5  2  9.5  0 


$2  2  5.5  0 

115.7  5 

9.7  5 

15.4  5 


$420.90 


Balance  due  E.  G.  J.  ^  Co.  $  1  0  8.6  0 

Received  Payment, 

E.  C.  Johnson  &Co. 


Miscellaneous  Examples  in  U.  S.  Money. 

1.  If  4  cords  of  wood  cost  $  34.50,  what  is  the  price  per 
cord?  Ans.  $8.62i 

2.  What  shall  I  pay  for  7  tons  of  hay,  at  $  16.75  per 
ton?  Ans.  $117.25. 

3.  My  farm  cost  $  3476.50  and  my  house  cost  $  2347.75  ; 
how  much  more  did  I  pay  for  the  farm  than  for  the  house  ? 

4.  When  beef  costs  12  J  cents  per  pound,  what  shall  I  pay 
for  1936  pounds?  '  Ans.  $242. 

5.  Bought  4  lb.  tea  at  75c.,  6  yd.  sheeting  at  33|c.,  and 
5yd.  broad  cloth  at  $  3.25  ;  what  was  the  cost  of  all  ? 

Ans.  $21.25. 

6.  If  8  yards  of  cloth  cost  $  12,  what  will  12  yards  cost? 

7.  If  16  barrels  of  flour  cost  $  144,  what  will  12  barrels 
cost?  Ans.  $108. 


148  UNITED    STATES   MONEY. 

8.  If  4  tons  of  coal  cost  $  35,  what  will  64  tons  cost? 

9.  If  4  tons  of  hay  cost  $  62.50,  what  will  48  tons  cost? 

10.  Paid  $  4050  for  75  acres  of  land  ;  at  what  price  per 
acre  shall  I  sell  it  to  gain  $  225  ?  Ans.  $  57. 

11.  Bought  22  pounds  of  sugar  at  9  cents,  4  pounds  of  cof- 
fee at  66  cents,  3  pounds  of  tea  at  75  cents,  5  gallons  of  mo- 
lasses at  48  cents,  and  5  barrels  of  flour  at  $  9.75,  and  gave 
the  merchant  6  ten-dollar  bills,  how  much  change  shall  he  return 
tome?  Ans.  $2.38. 

12.  Bought  6  pounds  of  butter  at  30  cents,  10  pounds  of 
cheese  at  18  cents,  24  pounds  of  rice  at  6  cents,  7  pounds  of 
raisins  at  25  cents,  2  bushels  of  potatoes  at  75  cents,  1  bushel 
of  beans  at  $  1.50,  and  10  yards  of  sheeting,  and  gave  2  ten- 
dollar  bills  to  the  merchant,  who  returned  $  7.91 ;  what  was 
the  price  per  yard  of  the  sheeting  ?  Ans.  23  cents. 

13.  A  merchant  bought  8 ''boxes  of  tea,  containing  60 
pounds  each,  for  $  312  ;  but  it  being  damaged  he  sold  it  at  a 
loss  of  $  72;  at  what  price  per  pound  did  he  sell  it?  How 
much  did  he  lose  on  each  pound  ? 

14.  A  family,  consisting  of  father,  mother,  and  2  children, 
desires  to  board  by  the  sea  during  the  summer,  and  can  afford 
to  pay  $  126  ;  how  many  weeks  can  they  remain,  if  the  board 
of  each  parent  costs  $5.50,  and  of  each  child  $3.50  per  week? 

Ans.  7. 

15.  A  laborer  bought  a  bushel  of  potatoes  for  75  cents,  6 
pounds  of  sugar  at  15  cents,  a  barrel  of  flour  for  $  8.75,  and 
12  pounds  of  meat  at  10  ceirts;  he  paid  $5.35  in  cash,  and 
the  balance  in  work  at  $  1.25  per  day ;  how  many  days  did  he 
work?  Ans,  5. 

16.  A  merchant  found  that  for  one  year  his  whole  profits 
were  $8750;  of  this  he  paid  $1250  for  store  rent,  $2750 
for  clerk  hire,  and  $  1850  for  other  expenses;  how  much  clear 
profit  remained  ? 


UNITED    STATES    MONET.  149 

17.  A  bookseller  went  to  the  city  and  bought  a  bill  of  books" 
as  follows  :  12  readers  at  $  1.25,  18  spellers  at  $.37^,  10  geo- 
graphies at  $  1.75,  8  primary  geographies  at  $.62^.  In  pay- 
ment he  gave  a  hundred  dollar  bill,  how  much  should  he  receive 
back? 

18.  It  took  25  yards  of  carpeting  at  8  1.87 J  per  yard  for 
my  sitting  room,  50  yards  of  matting  at  $.65  per  yard  for  my 
chambers,  and  39  yards  of  oil-cloth  at  $  1.25  per  yard  for  my 
kitchen  and  halls;  what  was  the  amount  of  my  bill? 

Ans.  $128.12^. 

19.  A  ship  carried  to  London  from  New  Orleans  4500  bales 
of  cotton,  each  bale  weighing  475  pounds,  at  a  freight  of  2^ 
cents  per  pound,  and  other  merchandise  upon  which  the  freight 
was  $  8595  ;  what  was  the  whole  amount  of  the  ship's  freight? 

20.  A  drover  bought  stock  as  follows :  7  horses,  at  an  average 
price  of  $  225  ;  50  sheep  at  $  3.50;  and  one  pair  of  oxen  for 
$200 ;  what  did  the  whole  cost  him? 

21.  A  gentleman  found  that  his  household  expenses  for  one 
month  were  as  follows :  provisions  $175.50,  groceries  $150.50, 
house  rent  $83.33  ;  what  was  the  amount?  What  would  be 
the  amount  for  one  year  ? 

22.  A  farmer  sold  the  produce  of  his  farm  as  follows:  150 
bushels  of  potatoes  at  $.55,  175  bushels  of  com  at  $1.25, 
and  50  bushels  of  wheat  at  $4.50  per  bushel;  what  was 
the  amount  he  received  ? 

23.  A  builder  took  a  contract  to  build  a  house ;  he  paid  for 
brick  and  stone  work,  with  materials,  $  3500 ;  for  carpenter 
work,  with  materials,  $2575.50;  for  painting  $675,  and  for 
other  work  $1550;  he  received  $10,000;  how  much  were  his 
profits  ? 


150  COMPOUND  NUMBERS, 


COMPOUND      NUMBEES. 

ADDITION. 

179.  A  Compound  Number  is  composed  of  two  or  more 
denominations  (Art.  92)  "whicli  do  not  usually  increase  deci- 
mally from  right  to  left ;  consequently,  in  adding  the  diflferent 
denominations,  we  do  not  carry  one  for  ten,  hut  for  the  numher 
it  takes  of  the  particular  denomination  added,  to  make  a  unit 
of  the  next  higher  denomination  ;  thus,  in  adding  Sterling  or 
English  money,  we  carry  1  for  4,  12,  and  20,  because  4qr. 
make  Id.,  12d.  make  Is.,  and  20s.  make  1£. 

Ex.  1.  Add  together  5£  10s.  7d.  3qr.,  6£  18s.  lid.  2qr., 
9£  13s.  5d.  Iqr.,  17£  16s.  9d.  3qr. 

We  first  arrange  the  numbers 
as  in  the  margin.  Then  add 
the  right-hand  column  as  in 
simple  numbers,  and  find  the 
amount  to  be  9qr.  :=  2d.  and 
3  9     19      10      1  Iqr.     We  write  the  Iqr.  under 

the  column  of  farthings,  and  add  the  2d.  to  the  column  of 
pence ;  the  amount  of  which  we  find  to  be  34d.  =  2s.  and 
lOd.  We  set  the  lOd.  under  the  column  of  pence,  and  add 
the  2s.  to  the  column  of  shillings,  and  find  the  amount  to  be 
593.  =:z2£  and  19s.  We  write  the  19s.  in  the  column  of 
shillings,  and  add  the  2£  to  the  column  of  pounds;  the 
amount  of  which  we  find  to  be  39£,  and  the  whole  amount 

£  8.  d.  qr. 

3  9       19       10  1  Ans. 

179,    What  is  a  compound  number  ?    How  do  they  increase  ?    What  is  said 
of  carrying  ? 


OPERATION. 

£ 

s. 

d. 

qr. 

0 

10 

7 

3 

6 

18 

11 

2 

9 

13 

5 

1 

1  7 

16 

9 

3 

ADDITION.  151 

180.  The  principle  of  this  process  is  precisely  the  same 
as  in  addition  of  simple  numbers.     Hence, 

To  add  compound  numbers, 

EuLE.  Write  the  numbers  so  that  each  denomination  shall 
occupy  a  separate  column,  the  lowest  denomination  at  the  rights 
and  the  others  towards  the  left  in  the  order  of  their  values.  Add 
the  numbers  in  the  lowest  denomination,  divide  the  amount  by 
the  number  it  takes  of  this  denomination  to  make  one  of  the  next 
higher,  set  the  remainder  under  the  column,  and  carry  the  quo- 
tient to  the  next  column.  So  proceed  until  all  the  columns  are 
added. 

Proof.     The  same  as  in  Addition  of  Simple  Numbers 

2.  3. 


£ 

s. 

d. 

gal. 

qt. 

pt, 

27 

17 

e 

5 

3 

19 

15 

10 

4 

2 

14 

6 

1 1 

7 

3 

28 

19 

9 

4 

0 

Sum, 

91 

0 

0 

22 

1 

Proof, 

91 

0 

0 

22 

1 

NoTK.  In  writing,  and  also  in  adding  the  numbers  of  a  single 
DENOMINATION,  the  Tules  of  simple  addition  must  be  observed ;  thu» 
in  writing  the  pounds  in  Ex.  2,  set  imits  under  units,  tens  under  tens. 


4. 

5. 

lbs. 

oz. 

dwt. 

grs. 

A. 

R. 

rd. 

17 

10 

19 

23 

7 

3 

27 

13 

7 

13 

19 

2 

2 

31 

7 

11 

17 

21 

6 

3 

28 

27 

10 

15 

20 

9 

3 

39 

Sum,   6  7 

5 

7 

1  1 

27 

2 

5 

Proof,  6  7 

5 

7 

11 

27 

2 

6 

152  COMPOUND    NUMBERS. 


6. 

7. 

bush.  pk.  qt.  pt. 

t. 

cwt.  qr. 

lb. 

oz. 

71  3  7   1 

7  ■ 

■19   3 

20 

13 

19  2  5   1 

5 

1  4   2 

16 

14 

3  3  0   0 

3 

17   3 

23 

2 

13  2  4   1 

4 

1  6   1 

19 

8 

8. 

9. 

yd.  qr.  na.  in. 

^rd.   yd. 

,  ft. 

in. 

5   3  3  2 

7   4 

2 

10 

7   2  3  U 

1   5 

2 

11 

9   3  2  2 

6   3 

1 

■  7 

7   3  3  2 

4   4 

2 

9 

21   2 

il 

I 

or  21      2     2        7 

Note.  A  fraction  occurring  in  the  amount  may  sometimes  be 
reduced  to  whole  numbers  of  lower  denominations ;  thus,  in  Ex.  9  ; 
we  reduce  the  iyd.  to  lower  denominations  =  1ft.  6in.,  this 
we  add  to  the  ft.  and  in.  in  the  example,  and  have  21rd.  2yd. 
2ft.  Tin. 

10.  A  trader  bought  4  hhd.  of  sugar:  the  first  weighed 
lOcwt.  3qr..  171b.;  the  second  13 cwt.  Iqr.  191b.;  the  third 
12cwt.  3qr.  18lb.;and  the  fourth  llcwt.  3qr.  271b.;  what  did 
the  whole  weigh?  Ans.  2t.  9cwt.  Iqr.  61b. 

11.  I  have  my  winter's  wood  in  four  piles;  in  one  are  4c. 
5  c.  ft.  12  cu.  ft;  in  another  2  c.  7  c.  ft.  9  cu.  ft.;  in 
another  1  c.  6  c.  ft.  13  cu.  ft.  and  in  the  fourth  3  c.  5  c.  ft.  11 
cu.  ft.;  how  much  wood  have  I  in  all? 

Ans.  13  c.  1  c.  ft.  13  cu.  ft. 

12.  A  vintner  has  wine  in  3  casks; in  the  first,  68gal.  3qt. 
Ipt.  3gi.;  in  the  second,  79gal.  2qt.  Ipt.  Igi.;  in  the  third, 
94gal.  3qt.  Ipt.  3gi.;  how  much  has  he  in  the  three  casks? 

180.  Rule  for  addition  of  compound  numbers?  Principle?  Proof? 
Numbere  of  a  single  denomination,  how  written  and  added  ? 


SUBTRACTION.  153 


SUBTKACTION. 


181.  The  principle  is  like  that  of  subtraction  of  simple 
numbers.     Hence, 

To  subtract  compound  numbers, 

EuLE.  1 .  Write  the  less  quantity  under  the  greater,  arrang- 
ing the  denominations  as  in  addition. 

2.  Beginning  at  the  right,  take  each  denomination  of  the 
subtrahend  from  the  number  above  it,  and  set  the  remxiinder 
beneath. 

3.  If  any  number  of  the  subtrahend  is  greater  than  the 
number  above  it,  add  to  the  upper  number  as  many  as  it  takes 
of  that  denomination  to  make  one  of  the  next  higher,  and  take 
the  number  in  the  subtrahend  from  the  sum  ;  set  down  thQ  re- 
mainder, and  considering  the  number  in  the  next  denomination 
in  the  minuend  one  less,  or  that  in  the  subtrahend  one 
GREATER,  proceed  as  before. 

Proof.  As  in  subtraction  of  simple  numbers, 

Ex.  1.  From  12£.  9s.  6d.  3qr.  take  8£.  7s.  9d.  2qr. 

We  take  2qr.  from  3qr.  and 
have  Iqr.  remaining,  which  we 
write  under  the  qr.  in  the  sub- 
trahend. We  see  that  we  cannot 
take  9d.  from  6d.,  we  therefore 
borrow  one  from  the  9  shillings, 
and  reduce  it  to  pence,  which  with  the  6d.  in  the  examples 
ISd,  We  now  say  9d.  from  18d.  leave  9d.  which  we  write  in  its 
proper  place,  under  the  pence  in  the  example.     Now,  as  one 

181.    Rule  for  subtraction  of  compound  numbers  ?   Principle  ?   Proof? 


OPERATION. 

£      s.     d.  qr. 
12     9      6     3 

8     7      9     2 

Sum, 

4     19     1 

Proof, 

12     9      6     3 

154  COMPOUND   NUMBERS. 

of  the  shillings  has  been  borrowed,  we  say,  7d.  from  8d.,  or 
what  is  practically  the  same,  8d.  from  9d.  leave  Id ,  and  so 
proceed  through  the  example, 

2.  From  S£  5s.  7d.  Iqr.,  take  3£  12s.  4d.  3qr. 

Ans.  4£  13s.  2d.  2far. 


3. 

4. 

t.  cwt. 

qr. 

lb. 

lb.  oz.  dr.  sc. 

grs. 

Min., 

19   12 

1 

20 

13  5   3  1 

10 

Sub., 

13  17 

3 

22 

7  9   12 

17 

Kern., 

5  14 
5. 

1 

23 

5  8  11 
6. 

13 

yd.  qr. 

na. 

gal.  qt.  pt. 

9   1 

2 

29  1  1 

3   3 

3 

13  3  1 

7. 

8. 

lb.   oz. 

dwt. 

gr- 

mi.  fur.  rd.  yd. 

ft. 

19   6 

1  2 

10 

8  2   21  2 

1 

12  10 

1  7 

21 

3  7   33  3 

2 

4     2      2  7    3^  2 

4    2      2  7    4     0     6  in. 


Note.  A  fraction  occurring  in  the  answer,  may,  when  reduced, 
contain  a  denomination  higher  or  lower  than  any  in  the  given  exam- 
ple ;  as  in  Ex.  8,  the  iyd.  =  1ft.  Gin.  The  1ft.  added  to  the  2ft.  in 
the  first  remainder  =  3ft.  =  1yd.  Add  this  to  the  3yds.  and  we 
have  4:mi.  2fur.  27rd.  4yd.  Oft.  Gin. 

Explain  Ex.  1. 


SUBTRACTION.  155 

10. 
deg.  mi.  fur.  rd.  yd.  ft.  in. 
'643321114 
3     62     5     37     1     2     7 


11.  If  I  cut  loyd.  oqr.  2na.  from  a  piece  of  clotli  con- 
taining 31yd.  2qr.,  bow  much  will  remain? 

Ans.  15yd.  2qr.  2na. 

12.  A  grocer  had  a  box  of  sugar  containing  15cwt.  Iqr. 
131bs.  After  taking  out  9cwt.  3c[r.  211bs.,  bow  much  remained 
in  the  box?  Ans.  5cwt.  Iqr.  171bs. 

13.  An  invoice  of  broadcloth,  which  cost  187£  17s.  6d., 
was  sold  for  257£  9s.  3d.;  what  was  the  gain? 

Ans.  69£  lis.  9d. 

14.  What  is  the  difference  in  the  longitude  of  two  places, 
one  63°  30'  15"  east,  and  the  other  23°  45'  30"  east? 

18S.     To  find  the  time  between  two  dates. 

Ex.  1.  What  is  the  difference  of  time  between  June,  11, 
1856,  and  Oct.  4,  1859? 

OPERATION.  In  subtracting  an 

Min.,         18  5  9         10  4  earlier  from  a  later 

Sub.,         1856  6         11  date,  we    call    30 

Kem.,  3y^        3  mo.  2  3  Ans.  days  a  month.  We 

write  first,  the  number  of  the  year,  month,  and  day  of  the  latest 
date,  and  under  it,  the  number  of  the  year,  month,  and  day  of 
the  earliest  date,  and  subtract  as  in  Art.  181,  and  the  re- 
mainder will  be  the  difference  of  time  between  the  two  dates. 

2.  Find  the  time  from  Sept  23,  1862,  to  May  13,  1866. 

Ans.  3y.  7mo.  20d. 

3.  Find  the  time  from  Aug.  17,  1858,  to  June  11, 1863. 

Ans.  4y.  9  mo.  24d. 


156  COMPOUND   NUMBERS. 

4.  Find  the  time  from  Feb.  8,  1856,  to  Aug.  1,  1860. 

5.  Find  the  time  from  March  7,  1857,  to  Nov.  20,  1865. 

MULTIPLICATION. 

183.  In  the  multiplication  of  both  simple  and  compound 
numbers,  the  multiplier  is  always  a  simple  abstract  number. 
The  product  is  of  the  same  kind  as  the  multiplicand ;  for  re- 
peating a  number  does  not  change  its  nature. 

The  principle  is  the  same  as  in  multiplication  of  simple 
numbers.     Hence, 

To  multiply  a  compound  by  a  simple  number  vre  have 
the  following 

Rule.  Multiply  the  lowest  denomination  in  the  multiplicand, 
divide  the  product  hy  the  number  it  takes  of  that  denoinination 
to  make  one  of  the  next  higher,  set  down  the  remainder,  add 
the  quotient  to  the  product  of  the  next  denomination,  and  so 
proceed  till  all  the  denominations  are  multiplied. 

Peoof.  3Iultiplication  and  Division  of  Compound  Numhers 
prove  each  other, 

,  2o[r.  by  9. 

We  first  say,  9  times  2qr.  = 
1 8qr.  =  4d.  and  2qr. ;  write  the 
2qr.  under  the  farthings,  and 
then  say  9  times  3d.  =  27d.  and 
the  4d.  added  give  31d.  :=  2  s. 
and  7d.,  and  so  proceed. 
Note.  As  raultiplication  and  division  prove  each  other,  it  is  profit- 
able to  teach  the  reverse  operations  simultaneously. 

183.  How  is  the  time  between  the  two  dates  found  ?  183.  What  kind  of 
a  number  is  the  multiplier  in  all  cases?  What  the  product?  The  rule  for 
multiplying  a  compound  number  ?    Proof?    Explain  Ex.  1. 


Ex.  1 

.  Multiply  7£  63.  3d, 

OPERATION. 

£     s. 
7      6 

d.    qr. 

3     2  Multiplicand. 
9  Multiplier- 

65  16 

7     2  Product. 

2 

Multiply 
By  • 

rd.  yd. 
9     4 

ft. 
2 

in. 

7 
8 

Product,  7  8     5 

2 

8 

4. 

lbs.  oz.  dwt. 

grs. 

6     7     13 

17 
5 

33     2        8 

13 

6. 

yd.  qr.  na. 
7      2      3 

in. 
2 

8 

8. 

gal.  qt.  pt. 
9       2      1 

gi- 

3 

.  3 

MULTIPLICATION.  157 


3. 

gal.  qts.  pt.  gi. 

9       3       13 

7 


69 

3      0     1 

5. 

lb. 

oz.  dr.  so. 

grs. 

3 

9     6     2 

14 

6 

. 

7. 

o            / 

It 

17     30 

45 

5 

9. 

bush.  pk.  qt. 
9       3      6 

pt. 

1 

11 

10.  How  much  vinegar  in  6  casks,  each  holding  37gal.  Iqt. 
Ipt.  3gi.  each?  Ans.  224gal.  3qt.  Opt.  2gi. 

11.  What  will  be  the  weight  of  3  loads  of  coal,  if  one  load 
weighs  1  ton,  Scwt.  3qr.  2 7 lbs.  (long  ton)  ? 

12.  What  is  the  entire  produce  of  a  field  of  8  acres,  if  one 
acre  produce  38bu.  3pk.  6qts.  ?  Ans.  311bu.  Ipk. 

13.  If  the  moon's  daily  motion  through  the  heavens  is  33° 
10'  35",  how  much  of  her  orbit  does  she  traverse  in  17  days  ? 

14.  If  a  horse  travel  42mi.  3fur.  37rd.  in  one  day,  how  far 
will  he  travel  in  16  days? 


1,58  COMPOUND   NUMBERS. 


DIVISION. 


184:.  Here,  as  in  the  three  preceding  sections,  the  prin- 
ciple is  the  same  as  in  the  corresponding  operation  in  simple 
numbers.     Hence, 

To  divide  a  compound  number  we  have  the  following 

BuLE.  Divide  the  highest  denomination  of  the  dividend, 
and  set  down  the  quotient ;  if  there  is  a  remaindery  reduce  it 
to  the  next  lower  denomination  ;  to  the  result  add  the  given  nmn- 
her  of  that  denomination^  and  divide  as  hefore,  setting  down  the 
quotient  and  reducing  ihe  remainder j  and  so  proceed  till  all  the 
denominations  are  divided. 

Proof.     Division  is  proved  hy' multiplication. 

Ex.  1.  Divide  27£  15s.  6d.  3qr.  by  8. 

OPERATION.  •  "VVe  first  divide  27£  by  8  and  have 

£  s.  d.  qr.  a  quotient  of  3,  and  3£  remaining. 
8)27  15  6  3  3£  reduced  to  shillings,  with  the  15 
3  9  5  1|  shillings  in  the  example,  give  75  shil- 
lings, which  divided  by  8,  give  9  and  a  remainder  of  3s. 
This  we  reduce  to  pence,  and  add  the  6d.  in  the  example,  and 
have  42d.,  which  we  divide  as  before,  etc. 

2.  Divides?  tons,  15cwt.  Iqr.  211b.  by  12. 

Ans.  3  tons,  2cwt.  3qr.  20ilb. 

3.  Divide  3 5y.  3mo.  17da.  13h.  by  3. 

4.  Divide  76a.  2r.  25rd.  by  5.         Ans.  15a.  Ir.  13rd. 

5.  If  5  loads  of  wood  contain  9c.  7c.  ft.  lOcu.  ft.,  what  are 
the  contents  of  1  load  ? 

6.  How  far  will  a  man  travel  in  one  day,  if  he  travel  1 71mi. 
Ifur.  29rd.  in  7  days?  Ans.  24mi.  3fur.  27rd. 

7.  If  it  take  250yd.  3qr.  2na.  of  carpeting  to  carpet  nine 

184.    Rule  for  dividing  a  compound  number  ?    Principle?    Proof? 


DIVISION".  159 

rooms,  how  many  yards  will  it  take  to  carpet  one  of  the  floors, 
they  being  of  equal  siz*  ?  Ans.  27yd.  3qr.  2na: 

8.  If  4doz.  spoons  weigh  6lbs.  lOoz.  16dwt.,  what  will  one 
dozen  weigh?  Ans.  lib.  8oz.  14dwt. 

9.  A  farmer  put  his  wheat,  consisting  of  359bu.  3pk.  2qt., 
into  12  boxes  of  equal  size ;  how  much  did  each  box  contain? 

Ans.  29bu.  3pk.  7qt.  Ipt. 

10.  If24hhd.  of  sugar  weigh  4t.  14cwt.  3qr.  51b.,  what 
is  the  weight  of  Ihhd.  ?  Ans.  3cwt.  3qr.  201b. 

11.  A  farmer  divided  his  farm  consisting  of  446a.  3r. 
30  rd.,  equally  among  his  8  children;  what  was  the  share  of 
each? 

Miscellaneous   Examples. 

1.  A  blacksmith  bought  5cwt.  2qr.  211b.  of  iron  at  one 
time,  It.  1  Icwt.  181b.  at  another ;  how  much  did  he  buy  in  all? 

Ans.  It.  16cwt.  3qr.  141b. 

2.  How  many  pounds  of  iron  did  the  above  blacksmith  buy, 
and  what  did  it  cost  him  at  5  cents  per  pound  ? 

3.  A  farmer  raised  in  one  field  302bu.  2pk.  7qt.  of  oats, 
in  another  290bu.  3pk.  4  qt. ;  how  much  more  did  he  raise  in 
one  than  the  other?  Ans.  llbu.  3  pk.  3  qt 

4.  I  have  a  piece  of  land  containing  50a.  ;  if  I  sell  25a. 
3r.  25rd.  of  it, how  much  shall  I  have  left? 

5.  If  a  ship  sail  2°  2'  30"  in  one  day,  how  far  will  she 
sail  in  a  week?  Ans.  14°  17' 30". 

6.  How  much  wood  in  5  loads,  each  containing  Ic.  3c.  fb. 
16cu.  ft.  ? 

7.  What  would  be  the  crop  of  hay  on  10a.  if  the  product  of 
1a.  was  3t.  10  cwt.  2  qr.  ?  Ans.  35  t.  5  cwt 

8.  Divide  2t  7  cwt.  2qr.  10  lb.  by  7. 


160  PERCENTAGE. 

9.  If  1  cubic  yard  of  stone  weigh  2  tons,    7cwt.   251K, 
what  is  the  weight  of  1  cubic  foot?       •  Ans.  1751b. 


PERCENTAGE. 


185.  The  term  Per  Cent  means  by  the  hundred ; 
thus,  by  jive  per  cent  of  a  ton  of  coal,  we  mean  five  one  hun- 
dredths of  it ;  i.  e.  five  parts  out  of  every  one  hundred  parts ; 
6  per  cent  of  a  sum  of  money,  is  six  one-hundredths  of  the  sum, 
i.  e.  $6  out  of  every  $100. 

Note.  Instead  of  the  WGrds  per  cent  it  is  quite  customary  in 
writing  to  use  the  sign  %\  thus,  6  per  cent  is  written  6  ^;  4i  per 
cent  4:i  %. 

186.  The  Rate  per  cent  is  the  number  far  each  hun- 
dred; thus,  6  Gjo  is  ToTT»  or  .06,  i.  e.  6  parts  for  each  hundred 


18T.  The  Percentage  is  the  sum  computed  on  the  given 
number;  thus,  the  percentage  on  %  200  at  6  per  cent  is  $  12. 

Note.  The  pupil  should  be  cautioned  not  to  confound  j»cr  cent 
and  'percentage.     The  distinction  should  clearly  be  borne  in  mind. 

188.  The  Base  of  percentage  is  the  number  on  which 
the  percentage  is  computed ;  thus,  we  say  the  percentage  on 
%  500,  at  8  per  cent  is  %  40.  Here  $  500  is  the  base,  8  is 
the  PER  CENT  and  $  40  is  the  percentage  ;  also,  10  per  cent, 
of  2000  lb.  (a  ton)  of  coal  is  2001b. ;  here  2000  lb.  is  the 
hasey  10  is  the  j»er  cent  and  2001b.  is  the  'percentage. 

185.  Meaning  of  per  cent  ?  186.  Kate  per  cent  ?  Illustrate.  187. 
Percentage?  Illustrate.  188.  Base  of  percentage  ?  Explain  the  three  last 
mentioned  terms  by  an  example. 


PERCENTAGE.  161 

189.  The  rate  per  cent  being  a  certain  number  of  hun- 
dredths, may  be  expressed  either  decimally,  or  by  a  common 
fraction,  as  in  the  following 

TABLE. 
Decimals.  Common  Fractions. 

1  per  cent        is         .01         =  t^^* 

2  per  cent  .02  =:  3^, 
5  per  cent  .05  =  ^^, 
6i  per  cent  .0625  —  yV 
8i  per  cent                   .08^       =:             yV- 

10    per  cent  .10  =:  yV 

12i  per  cent  .125  —  \. 

16f  per  cent  .16|  =  f 

181  per  cent  .1875  —  yV 

20    percent  .20  =  |. 

25    per  cent  .25  =  \. 

33i  per  cent  .33^  =  \. 

60    per  cent  .50  =  i. 
etc.                            etc. 

Note.  When  the  per  cent  is  expressed  by  a  decimal  of  more 
than  2  places,  the  figures  after  the  second  decimal  place  must  be 
regarded  as  parts  of  1  per  cent ;  thus,  (in  the  seventh  line  of  the  fore- 
going table,)  .125  is  12i^  or  12^  per  cent. 

Ex.  1.  Write  the  decimal  for  6  per  cent.  Ans.  .06. 

2.  Write  the  decimal  for  4  per  cent;   12  per  cent;   8  per 
cent;  15  per  cent;  25  percent;  16J  per  cent. 

3.  Write  the  common  fraction  for 5  per  cent;  10 per  cent; 
12 J  per  cent ;  6i  per  cent;  33 J  per  cent. 


189.    In  what  ways  may  the  rate  be  expressed  ?    If  expressed  decimally 
by  more  than  two  figures,  what  are  the  figures  after  the  second  decimal  place  ? 


162  PERCENTAGE. 

Note.     Too  much  pains  can  not  be   taken  to  make  the  pupil 
thorough  in  exercises  like  those  in  the  last  two  examples. 


Case  1. 

190.  To  find  the  percentage,  the  base  and  rate  per 
cent  being  given. 

Ex.  1.  John  Dow  had  $  360,  hut  lost  5  per  cent  of  it,  how 
many  dollars  did  he  lose  ? 

$  3  6  0  Since  5  per  cent  is  .05  =  ^i^,  we  find  the  loss, 

'Q  ^     (percentage),  by  multiplying  $360  by  .05  or  by 
$  1  8.0  0     s^j.     Hence,  the 

KuLE.  3Iultiplt/  the  base  hy  the  rate  'per  cent  expressed 
decimally  or  as  a  common  fraction,  and  the  product  will  he 
the  percentage. 

2.  What  is  20  per  cent  of  $  IGO.  ?  Ans.  $  32. 

3.  The  base  is  560  and  the  rate  per  cent  40 ;  what  is  the 
percentage  ? 

560  X  .40  =  224,  Ans. 
Or,     560  X    f  =  224,  Ans. 

4r.  What  is  16|  per  cent  of  180  barrels  of  flour? 

Ans.  30  bbl. 

5.  What  is  5  %  of  $  200  ?  Ans.  %  10. 

6.  What  is  8i  %  of  240  tons  of  coal?       Ans.  20  tons. 

7.  In  a  certain  school  there  are  720  pupils,  33|  per  cent 
are  more  than  12  years  of  age;  how  many  are  over  12 
years  old  ?  Ans.  240. 

190.    Rule  for  finding  the  percentage  when  the  base  and  rate  are  given  ? 


PERCENTAGE.  163 

8.  A  flour  merchant  bought  1200  bbls.  of  flour,  but  8  per 
cent  of  it  was  injured  by  rain ;  how  much  was  injured? 

9.  A  pupil  had  a  lesson  of  40  words,  but  failed  on  10  per 
cent  of  them  ;  on  how  many  words  did  he  fail  ? 

10.  A  city  containing  35000  inhabitants,  had  15  per  cent 
of  the  number  in  school  children ;  how  many  school  children 
were  there  ?  An s.  5250. 

11.  A  merchant  fails  in  business,  owing  $  12G00,  and  can 
pay  but  35  per  cent  of  his  debts ;  how  much  will  his  credi- 
tors lose  ? 

100  —  35       =65 

12600  X. 6  5  =$8  19  0,  Ans. 

12.  Bought  600  boxes  of  oranges,  but  on  opening  them,  I 
find  8  %  of  them  spoiled ;  how  many  were  lost?     Ans.  48. 

13.  A  gentleman  sold  his  house  for  $6500.  $600  he  re- 
ceived in  cash  and  took  a  note  for  the  balance ;  how  much 
cash  did  he  receive  ? 

14.  I  have  $  1580  on  deposit  in  the  bank.  If  I  draw  out 
12  J  per  cent  of  it,  what  per  cent  will  remain  ?  What  amount 
of  money  will  remain?  Last  Ans.  $1382.50. 

15.  In  an  orchard  consisting  of  1200  trees,  30  %  bear 
apples,  45  %  bear  pears,  and  the  remainder  bear  peaches ;  how 
many  bear  peaches?  Ans.  300  trees. 

Case  2. 

191  •  To  find  the  rate  per  cent  when  the  base  and 
percentage  are  given. 

Ex.  1.  What  per  cent  of  8  24  is  $  6.  ? 

S  6  is  i  of  $  24   and  I  re^ 
2^i  =:  J  =  .2  5,     Ans.  duced  to  a  decimal  =  .25  L  e. 

$  6  =  25  per  cent  of  $  24. 
Hence  tho 


164  PERCENTAGE. 

Rule.  Mahe  the  'percentage  the  numerator  of  a  common 
fraction  and  the  base  the  denominator^  and  then  reduce  this 
fraction  to  a  decimal, 

2.  "What  per  cent  of  $  20  is  $  5  ?  Ans.  .25. 

3.  What  per  cent  of  $  400  is  ^  50  ?  Ans.  .12^-. 

4.  A  man  having  $  6000,  paid  %  1200  for  a  piece  of  land  ; 
what  per  cent  of  his  money  did  he  expend?  Ans.  .20. 

5.  My  salary  is  $1800  per  annum,  and  my  expenses 
$  1600  ;  what  per  cent  of  my  income  do  I  spend?  What  per 
cent  save  ? 

6.  Bought  a  cask  of  vinegar  containing  84  gallons;  21 
gallons  have  leaked  out ;  what  per  cent  have  I  lost  ? 

7.  Purchased  a  horse  for  $  160,  and  sold  him  for  $  128, 
what  per  cent  did  I  lose  ? 

Case   3. 
193.      To  find  the  base  when  the  percentage  and 
the  rate  are  given. 

Ex.  1.  $  12  is  4  per  cent  of  what  sum?  Ans.  |  300. 

If  $  12  is  4  per  cent,  1  per  cent  will  be  |  of  $  12  which  is 
I  3,  and  if  $  3  is  1  per  cent,  100  per  cent  will  he  100  times 
$  3  =:  S  300.  The  same  result  is  obtained  by  multiplying  by 
100  first,  and  then  dividing  by  4;  thus,  1200-^4  =  300. 
Hence, 

EuLE.  Multiply  the  percentage  by  100,  and  divide  the  prod- 
uct by  the  rate,  and  the  quotient  will  he  the  base. 

2.  $  12.60  is  6  %  of  what  sum?  Ans.  $  210. 

3.  $  15  is  8  %  of  what  sum?  Ans.  $  187.50.   ' 

191.  Rule  for  finding  the  rate  per  cent  when  the  base  and  percentage  are 
given  ?  19'^.  Kule  for  finding  the  base  when  the  percentage  and  rate  are 
given  ? 


INTEREST.  165 

4.  $36.30  is  3  %  of  what  sum? 

5.  $  75  is  6  %  of  what  sum  ?  Ans.  $  1250. 

6.  $  12  is  4  %  of  what  sum  ?  Ans.  $  300. 

7.  A  gentleman  purchased  a  farm  for  $  6900,  which  was 
20  per  cent  of  his  entire  property.     What  was  he  worth  ? 

8.  James  Marvin  cures  fish  for  Thomas  Tarlton,  receiv- 
ing in  pay  12^  per  cent  of  the  quantity  cured.  His  share  this 
season  is  46801bs ;  what  quantity  did  he  cure? 

9.  A  merchant  saves  $3000  annually,  which  is  16|  per 
cent  of  his  entire  receipts;  what  arc  his  receipts? 

Ans.  $18000. 
10.  A  farmer  sold  56  sheep,  which  was  12^  per  cent  of  his 
whole  flock.     How  many  sheep  had  ho  in  all  ?      Ans.  448. 

INTEEEST. 

103*     Interest  is  money  paidybr  the  use  of  money. 

The  Principal  is  the  sum  for  which  interest  is  paid. 

The  Amount  is  the  sum  of  the  principal  and  interest. 

194:.  An  example  in  interest  is  only  a  question  in  per- 
centage. HhQ  principal  is  the  base  of  percentage,  the  interest 
is  the  percentage,  and  the  interest  on  $1  for  a  year  is  the  rate 
written  decimally. 

100.  The  rate  is  usually  Jixed  hy  law,  and  a  higher 
rate  than  the  law  allows  is  called  usury. 

In  New  England  and  most  of  the  United  States  the  legal 
or  lawful  rate  is  6  per  cent ;  in  New  York,  7  per  cent. 

Note.  In  Massachusetts  a  higher  rate  than  six  per  cent  may 
legally  be  agreed  upon.  In  this  book  6  per  cent  is  understood 
when  no  per  cent  is  mentioned. 

193.  What  is  Interest  ?  What  is  the  Principal  ?  What  is  the  Amount  f 
194.  What  is  said  of  an  example  in  Interest  ?  Explain  how  the  latter  is 
like  one  in  percentage.    195.    What  is  the  Rate  ?     The  legal  Rate  ? 


166 


PERCENTAGE. 


100.  To  find  the  interest  on  any  sum  at  6  per 
cent  for  a  given  time. 

Ex.  1 .  What  is  the  interest  of  $  240,  for  one  year  6 
months  ? 

OPERATION. 

^240.     Principah 

Interest  for  2  mo. 

Interest  for  1  mo. 
Months  z=  ly.  6 mo. 


2)2.4  0, 

1.2  0, 

1  8 

960 
120 


ANALYSIS. 

Any  sum  of  money  at 
6  %  will  gain  jwo  of  it- 
self in  2  months.  AVe 
find  this  hy  removing  the 
decimal  point  two  places 
to  the  left,  that  is,  by  di- 
^21.60,  Interest  for  18  mo.  viding  by  one  hundred. 
We  then  divide  this  interest  by  2  which  gives  the  interest  for 
one  month.  This,  we  multiply  by  the  number  expressing  the 
given  time  in  months,  and  have  the  interest  of  $  240  for  ly. 
6mo.  =  $21.60. 

Ex.  2.  What  is  the  interest  of  $  244.40  for  1  year  4  mo. 
12  days? 

OPERATION. 

$2  44.40,     Principal. 
2  )  2.4  4  4=r  Interest  for  2  mo. 
1.2  22  =  Interest  for  1 


mo. 
1  G.4  mo.  =  ly.  4mo.   12d. 


ANALYSIS. 

In    this    example   we 

proceed  as  in  Ex.  1,  till 

we    come   to   the  days. 

12  days  =  -J^  of  a  month, 

which   we    reduce   to   a 

decimal  =  .4  and  annex 

to  the    16    months,  and 

$2  0. 0408  =  Int.  for  IG^V  ^^-      multiply  as  before. 

Hence  to  find  the  interest  on  any  sum  for  any  time  at  6  per 

cent  we  have  the  following 

Rule.     Bemove   the  decimal  point  in   the  principal^  two 

places  to  the  left ;  divide  this  result  hy  two  and  multiply  this 

196.    How  do  you  find  the  interest  on  any  sum  for  any  time  at  6  per  cent  ? 
Expiaiu  the  stepe. 


4888 
T332 
1  222 


INTEREST.  167 

quotient  hy  the  time  in  months  and  tenths  of  a  month  {the  days 
divided  hy  ^  =z  tenths  of  a  month),  and  the  product  will  be 
the  interest  at  6  per  cent  for  the  given  time, 

Ex.  3.  What  is  the  interest  of  $  46  for  1  month? 

$46.  -^  100  =  46,  and  .46  -^  2  zrr  $  .23,  Ans. 

4.  What  is  the  interest  of  $  246.58  for  1  month? 

$  246.58  -^  100  =  2.4658,  -^  2  =  $  1.2329. 

5.  What  is  the  interest  of  S  4,  for  1  month  ? 

$  4  -^-  100  =:  .04,  -^2  =  .02,  Ans. 

6.  What  is  the  interest  of  $  1,  for  1  month? 

$  1.  _i-  100  =  .01,  -^2  —  .005,  Ans. 

7.  What  is  the  interest  of  $  56.^98,  for  1  month? 

8.  What  is  the  interest  of  $  864.25,  for  1  month? 

Ans.  $4.32125. 

9.  What  is  the  interest  of  $  69.42,  for  1  month  ? 

10.  What  is  the  interest  of  S  2468.20,  for  1  month? 

11.  What  is  the  interest  of  $59,278,  for  1  month? 

12.  By  what  will  you  multiply  the  interest  for  one  month 
to  find  the  interest  for  1  year,  4  months  ? 

ly.  4mo.  =:  16mo.  Ans.  16. 

13.  By  what  will  you  multiply  to  find  the  interest  for  2 
years,  7  months? 

14.  By  what  will  you  multiply  to  find  the  interest  for  15 
days?  15d.  = -^^mo.  =^mo.  =.5mo.  Ans.  .5. 

15.  By  what  will  you  multiply  to  find  the  interest  for  21 
days? 

16.  By  what  will  you  multiply  to  find  the  interest  for  22 
days?  22d.  =  ffmo.  =  .7jmo.  Ans.  .Zj. 

17.  By  what  will  you  multiply  to  find  the  interest  for  9 
months,  13  days?  9rao.  13d.  =  9|§mo. :;:=:  9.4jmo. 

Ans.  9.4J. 


168  PERCENTAGE. 

18.  By  what  will  you  multiply  to  find  the  interest  for  ly. 
7mo.  19d.? 

19.  By  what  will  you  multiply  to  find  the  interest  for  2y. 
9mo.  20d.  ? 

20.  What  is  the  interest  of  S  164.30,  for  18  days? 
164.30  -^  100  =  1.6430,  -^  2  z=  .8215,  X  .Q~$A929, 

21.  What  is  the  interest  of  $  58.64,  for  2y.  3mo.  21d.  ? 
$  58.64  -^  by  100  =  .5864,  -^  2  =  .2932,  X  27.7  = 

$8.12+. 

22.  Find  the  interest  on  $  619.28,  for  3y.  4mo.  26  days. 

Ans.  $132.66. 

23.  Find  the  interest  on  $  384.92,  for  2y.  5 mo.  27  days. 

24.  Find  the  interest  on  $  87.25,  for  ly.  Imo.  10  days. 

Ans.  $5,816. 

25.  What  is  the  amount  of  $  142.80,  for  ly.  6mo.  24  days? 

Ans.  $156,223. 

Note.     The   amount  is  the  sum  of  the  principal   and  interest 
added  together. 

26.  What  is  the  amount  of  $  234.60,  for  6y.  7mo.  8  days? 

Ans.  $  327.57+. 

27.  What  is  the  amount  of  $  104.20,  for  6mo.  19  days? 

Ans.  $  107,655+. 

28.  What  will  $  380.50  amount  to  in  ly.  5rao.  10  days? 

29.  Find  the  interest  of  $  60,  for  60  days.        Ans.  $  .60. 

30.  Find  the  interest  of  $  30,  for  90  days. 

31.  Find  the  interest  of  $240.60,  for  2y.  llmo.  28d. 

Ans.  8  43.227. 

32.  Find  the  amount  of  $  350,  for  3y.  7mo.  lOd. 

33.  Find  the  interest  on  $  3,  for  7  days.         Ans.  .0035. 

34.  Find  the  interest  on  $  .80  for  10  days.  Ans.  .0013i. 

35.  Find  the  interest  on  $2.42,  for  25  days. 


INTEREST.  169 

36.  Find  the  amount  of  $  5,  for  12  days.     Ans.  $  5.01. 

37.  rind  the  amount  of  $  75.60,  for  8mo.  29d. 

Ans.  $78,989. 

38.  Find  the  amount  of  $  3000,  for  2y.  8mo. 

39.  Find  the  amount  of  $  230,  for  7y.  6mo. 

Ans.  $333.50. 

40.  Find  the  interest  of  $  394.27,  for  8y.  23d. 

41.  Find  the  amount  of  $  6000,  for  9y.  7mo.  23d. 

197,  In  all  the  previous  examples,  the  interest  has  been 
computed  on  the  basis  of  6  per  cent.  In  Ex.  42,  we  first  cast 
the  interest  at  6  per  cent,  as  before,  and  find  it  to  be  $1 7.001  ; 
this  we  divide  by  6,  which  gives  the  interest  at  1  per  cent; 
and  lastly,  we  multiply  this  interest  by  7  and  have  the  interest 
at  7  per  cent  =  $  19.83+. 

42.  What  is  the  interest  of  $  1 64,  for  ly.  8mo.  22d.  at  7 
percent?  Ans.  $19.83+. 

43.  \Yhat  is  the  interest  of  $270.60,  for  3y.  llmo.  19d. 
at  8  per  cent?  Ans.  $  85.93. 

44.  What  is  the  interest  of  $  492.75,  for  2y.  7mo.  at  7i 
per  cent  ? 

45.  Find  the  interest  on  $  75.87,  for  5y.  3mo.  at  9  percent. 

Ans,  $  35.848. 

46.  Find  the  interest  on  $  894.20,  for  3y.  6mo.  15d.  at  8 
per  cent.  Ans.  S  253.35.+ 

47.  Find  the  amount  of  $382.85,  for  4y.  lOmo.  at  9  per 
cent. 

48.  Find  the  amount  of  $  69.47,  for  3y.  8mo.  at  6^  per 
cent.  Ans.  $  86.02. 

197.  How  is  interest  found  for  any  other  rate  than  6  per  cent  ?  Esplaia 
the  steps.    How  is  the  difference  of  time  between  two  dates  found  ? 


170  PERCENTAGE. 

49.  Find  the  interest  of  $609.42,  for  8y,  7mo.  6d.  at  4 
percent.  Ans.  $209.64. 

50.  Find  the  interest  on  $  493.85,  for  2y.  3mo.  at  3^  per 
cent.  Ans.  $38.89. 

51.  Find  the  interest  on  $6000,  for  87.  7mo.  17d.  at  SJ 
per  cent. 

52.  What  is  the  interest  of  $  64.82  from  June  24,  1856,  to 
>Oct.  9,  1859?  Ans.  $  12.80. 

Note.     First  find  the  difference  of  time  =  3y.  3rn.  15d. 

53.  What  is  the  interest  of  $  85.93  from  Jan.  6,  1850,  to 
June  1,  1854? 

54.  What  is  the  interest  of  $  942.87  from  Aug   13,  1861, 
to  Nov.  7,  1864?  Ans.  $182.91. 

55.  What  is  the  interest  of  $  293.80  from  Feb.  19,  1860, 
to  Sept.  4,  1863  ?  Ans.  $  62.43. 

bQ.  What  will  $  843.92  amount  to,  from  Aug.  28,  1862,  to 
Jan.  1,  1866?  Ans.  $1013.12. 

57.  What  is  the  interest  of  $  59.75  from  Dec.  29,  1858,  to 
June  7,  1861,  at  7t  per  cent  ?  Ans.  $  10.92. 

58.  What  will  be  the  amount  of  $  642.90  from   July  4, 
1862,  to  the  day  of  Gen.  Lee's  surrender? 

59.  Find  the  interest  of  $  8942,  from  the  fall  of  Fort 
Sumpter  to  the  evacuation  of  Eichmond  ? 


PEOFIT  AND  LOSS. 


198.  Profit  and  Loss,  are  commercial  terms,  used  to 
indicate  the  gain  or  loss  in  buying  and  selling  goods,  and  in 
business  transactions  generally. 

198.    "What  is  Profit  and  Loss  ? 


PROFIT   AND   LOSS.  171 


Case  1. 


199,  To  find  the  absolute  gain  or  loss  on  a  quan- 
tity of  goods  sold  at  retail,  the  purchase  price  of  the 
whole  quantity  being  given. 

Ex.  1.  Bought  2001b.  of  coffee  for  $  50  and  sold  it  af 
S  .40  per  lb.;  how  much  did  I  gain  on  the  whole  ? 

§  200.  X  .40  ==  $  80.;  80  —  50.  z=  80.        Ans.  |  30. 
Hence  the 

EuLE.  Find  the  whole  sum  received  for  the  goods,  and  the 
difference  between  this  and  the  purchase  price  will  he  the  gain 
or  loss. 

2.  -Bought  32  yards  of  cloth  for  %  48  and  sold  it  at  $  2.20 
per  yard;  what  was  my  whole  gain?  Ans.  S  22.40. 

3    Bought  4cwt.  3qr.   161b.  sugar  for  $61,375,  and  sold 
it  at  $  .20  per  lb.     How  much  did  I  make  on  the  whole  ? 

4.  Bought  960  oranges  for  %  19.20  and  sold  them  for  5 
cents  apiece  ;  what  was  my  entire  gain  ? 

5.  Bought  15  doz.  pencils  at  $  1.20  per  doz.  and  sold  them 
at  $.15  each;  what  did  I  make  ? 

Case  2. 

300.  To  find  the  per  cent  of  gain  or  loss  when 
the  cost  and  selling  price  are  given. 

Ex.  1.  Bought  12  yards  of  cloth  for  $  18.  and  sold  it  at 
$  2.50  per  yd.  What  was  my  whole  gain,  and  what  my  gain 
per  cent? 

199.    Rule  for  finding  absolute  gain  or  loss  ? 

aOO.    Kule  for  finding  the  per  cent  of  gain  or  loss  on  the  cost  price  ? 


172  PERCENTAGE. 

$  2.5  0    Selling  price.         "We  first  find  the  whole  gain  to  bo 

, L?  y^'^^^-  $  12  which  is  if  —  §  of  the   cost. 

3  0.0  0  =  Sum  rec'd.  This  we  reduce  to  a  decimal,  and 
IQ        =Cost.  have    .66|;    i.  e.,  the  gain  $12,  is 

$12  =  Whole  gain.  66 f  per  cent  of  the  cost  $  18.  Hence, 
J|  =  I  =3  .66|.       the  following. 

EuLE.  Having  found  the  total  gain  or  loss  hij  Case  1, 
make  a  common  fraction  hy  writing  the  gain  or  loss  for  the 
numerator  and  the  cost  of  the  article  for  the  denominator,  and 
then  reduce  this  fraction  to  a  decimal. 

2.  Bought  a  gross  of  steel  pens  for  $  1.20,  and  sold  them 
at  1  cent  apiece.     What  was  my  gain  per  cent  ? 

Ans.  20%. 

3.  I  purchased  1  bushel  of  cherries  for  $  3,  and  sold  them 
at  15  cents  per  quart.     What  was  my  gain  per  cent? 

4.  Sold  a  horse  for  $  100  which  cost  mo  $  150.     What 
was  my  loss  per  cent  ?  /   . 

Case  3. 

301.  To  find  the  selling  price,  the  cost  and  gain 
or  loss  per  cent  being  given. 

Ex.  1.  Bought  cloth  at  $  6  per  yard  and  wish  to  sell  it  at 
an  advance  of  20%.     What  is  my  selling  price? 

$  6.0  0  I  shall  sell  what  cost  me  $  1.  for  $  1.20 

1-2  0  and  what  cost  me  $  6  for  six  times  $  1.20 

$7.2  0  0  0,  Ans.     r=  7.20. 

Ex.  2.  Bought  goods  for  $  300,  but  they  being  damaged  I  am 
willing  to  sacrifice  15  %  of  their  value ;  for  what  shall  I  sell 
them? 

'^01.  Ilule  to  find  the  selling  price,  the  cost  and  gain  or  loss  per  cent  bein^j 
given  ? 


PROFIT  AND   LOSS.  173 

$  3  0  0.  If  I  lose  15%,  I  shall  sell  what 

•8  5  cost  me  S  1.  for  S. 85,  and  shall  sell 

15  0  0  what  cost  me  $  300.  for    300  times 

2400  $.85=8  255. 

$  2  5  5.0  0  Hence,  we  have  the  following 

Rule.  Multiply  the  cost  by  1  minus  the  loss  or  plus  the 
GAIN  PER  CENT,  and  the  product  will  be  the  selling  price. 

3.  For  what  must  I  sell  sugar  that  cost  me  15  cents  per 
pound,  to  gain  30%  ?  Ans.  $  .19^. 

4.  Bought  a  pair  of  skates  for  $  4.00 ;  for  how  much  must  I 
sell  them  to  gain  10%  ? 

Case  4. 

dO^.  To  find  the  first  cost  of  an  article,  the  selling 
price  and  gain  or  loss  per  cent  being  given. 

Ex.  1.  Sold  wine  at  $6  per  gal.  and  by  so  doing  I  made  20 
per  cent  on  the  cost ;  what  was  the  cost  ? 

That  which  cost  $  1.  sold  for  S  1.20, 
12.^  zzr  I  therefore,  the  cost  was  j^^o  z=  |.  of  the 

$6.Xf^=S5.=:  cost,    selling  price ;  henee,  the  cost  was  f 
of  $6.  =  $5. 
Hence  v^^e  have  the  following 

Eule.  Make  a  fraction  hy  writing  100 /or  tlw  numerator 
and  100  minus  the  loss  or  plus  the  gain  per  cent  for  a  denomi- 
nator, then  multiply  tlie  selling  price  by  this  fraction. 

2.  Flour  selling  at  S  15  perbbl.,  yields  a  profit  of  25%; 
what  is  the  cost?  Ans.  $  12, 

3.  Sold  8  yards  of  cassimere  at  $3  per  yd.,  and  made  20% 
by  the  sale;  what  was  the  cost?  Ans.  $2.50  per  yard. 

4.  Bought  a  quantity  of  wheat,  but  it  being  damaged,  I 
sell  it  at  %  1.50  per  bushel,  and  by  so  doing  lose  25  per  cent 
on  the  cost ;  what  was  the  cost  ?  Ans.  8  2.00. 

202.  Rule  for  findinjf  the  cost,  the  selling  price  and  gain  or  loss  being 
given  ? 


174  MISCELLANEOUS. 


MISCELLANEOUS     EXAMPLES. 

1.  Add  120  X  2  to  972-^36.  Ans.  267. 

2.  The   difference   between  two  numbers  is   16,  and   the 
larger  is  92  ;  what  is  the  smaller  ? 

3.  What   is  the  difference    between  2446352  4-694701 
and  2146705  —  8392-1:1? 

4.  What  is  the  difference  between  246001  X  1641   and 
245897  X  321  ?  Ans.  324754704. 

5.  Multiply  12948  by  287,  subtract   58672,  and  divide 
the  remainder  by  218. 

6.  How  many  cords  in  a  pile  of  wood  86  ft.  long,  4  ft. 
wide,  and  9  ft.  high?  Ans.  24^^^. 

7.  Divide  |  of  f  by  |  of  J-^.  Ans.  h 

8.  What  number  multiplied  by  8|  gives  205  ? 

9.  What  number  divided  by  19f  gives  36  ?    *Ans.  708. 

10.  At  S10§  per  ton,  what  will  be  the  cost  of  |  of  a  ton 
of  coal? 

11.  If  If  yards  of  cloth  are  required  for  1  coat,  how  many 
coats  may  be  made  from  2 2f  yards?  Ans.  13. 

12.  What  is  the  difference  between  eight  hundred  thousand 
and  eight  hundred-thousandths?  Ans.  799999.99992. 

13.  If  18   gentlemen  have   $646.70  each,  what  sum  have 
they  all?  Ans.  $11640.60. 

14.  Bought  19  bbls.  of  flour  for  $261.25;    what  was  the 
price  per  bbl.? 

15.  If  the  crop  of  hay  on  1  acre  is  1  ton,  18  cwt.  3  qr. 
15  lbs.  what  will  be  the  crop  on  10  acres?   Ans.  19  t.  9  cwt. 

16.  The  population  of  a  certain  city  is  29460;  what  will 
it  be  a  year  hence  if  it  gains  5  %  ? 


MISCELLANEOUS .  175 

17.  If  a  farmer  raises  2650  bushels  of  wheat  one  year  and 
2968  the  next ;  what  per  cent  did  his  crop  increase  ? 

Ans.  1 2  per  cent. 
.18.  A  man  with  a  salary  of  $1600,  spends  $1200  ;  what 
per  cent  of  his  salary  did  he  save  ? 

19.  What  is  the  interest  of  $1.00  for  8  yrs  8  mos.  and  6 
days?  Ans.  $0,521. 

20.  What  is  the  interest  of  $500.00  for  60  days,  at  7y3^%? 

21.  Find  the  interest  of  $460  for  2  yrs.  4  mos.  and  1 8  days. 

Ans.  $65.78. 

22.  What  is  the  interest  of  $84.75  for  10  yrs.  6  mos.  24 
days,  at  9%  ?  Ans.  $80,597. 

23.  The  credit  side  of  an  account  is  composed  of  the  fol- 
lowing items:  $1500.75,  $655.30,  $175,875;  what  is  the 
whole  amount  ? 

24.  The  debit  side  of  the  above  account  has  the  following 
items:  $576.37,  $1025.50,  and  $1850.00;  what  is  the  whole 
amount  ? 

25.  On  which  side  of  the  above  account  is  the  balance,  and 
how  much  is  it?  Ans.  debit;  $1119.945. 

26.  How  many  yards  of  cloth  at  $2.25  per  yard  should  be 
received  for  8  cords  of  wood  at  $6.20  per  cord  ? 

Ans.  22/^  yds. 

27.  In  1  week,  1  day,  16  hours,  and  40  min.  how  many 
seconds?  Ans.  751200. 

28.  Resolve  1820  into  its  prime  factors. 

29.  Add  f ,  |,  },  and  f  Ans.  2|f. 

30.  Bought  a  house-lot  120  ft.  long  and  90  ft.  wide,  at  10 
cts.  per  square  foot.  The  cost  of  the  lot  was  12%  of  the 
cost  of  the  house ;  what  was  the  cost  of  both  ? 

Ans.  $10080.00. 

31.  If  12  bbls.  flour  cost  $135.00,  what  will  50  bbls.  cost? 

32.  Divide  four  thousand  by  eight  thousandths. 


176  MISCELLANEOUS. 

33.  If  yS^  of  a  vessel  cost  ^3690,  what  will  f  cost  ? 

Ans.  $6396. 

34.  What  is  the  interest  of  $1245.60  from  Jan.  20,  1866, 
to  May  2,  1867? 

35.  The  population  of  the  New  England  States,  in  round 
numbers*  is  as  follows:  Maine  628000,  New  Hampshire 
326000,  Vermont  315000,  Massachusetts  1231000,  Rhode 
Island  175000,  Connecticut  460000  ;  what  is  the  entire 
population  of  New  England  ? 

36.  Bought  corn  at  $2.00  a  bag,  and  sold  it  for  $1.60  ; 
how  much  did  I  lose  per  cent  ?  Ans.  20%. 

37.  If  1  ton  of  coal  costs  $7.75,  how  many  tons  can  be 
bought  for  $147.25? 

38.  Bought  a  pile  of  wood  40  ft.  long,  12  ft  wide,  and  15 
ft.  high,  at  $6.00  a  cord  ;  how  many  cords  were  there,  and 
what  was  the  expense  ?  Ans.  56  J  cords,  and  $337.50. 

39.  Eind  the  greatest  common  divisor  of  1504  and  3478. 

40.  Eind  the  least  common  multiple  of  6,  12,  18,  36, 
and  54. 

41.  If  it  cost  $25.00  for  1000  miles  travel,  what  is  that  a 
mile  ?  Ans.  2|  cents. 

42.  Bought  a  house  for  $1800.00,  which  was  f  of  what  I 
paid  for  my  farm ;  what  was  the  cost  of  the  farm  ? 

Ans.  $4200.00, 

43.  A  manufacturer  sends  34  cases  of  shoes  to  a  com- 
mission merchant.  These  are  sold  at  $140  a  case.  The 
commission  is  2i  per  cent.  What  is  the  whole  amount  of 
the  commission  received,  and  how  much  is  due  the  manu- 
facturer ? 

44.  Multiply  25893  by  .000402.  Ans.  10.408986. 

-  45.  If  I  give  .35  of  a  cord  of  wood  for  1  day's  work,  how 
much  should  I  give  for  64.50  days'  work?  How  much  would 
it  be  worth  at  $5.00  a  cord? 


MISCELLANEOUS.  177 

46.  What  is  the  interest  of  $800.50  from  June  19,  1860, 
to  Nov.  4,  1865  ?  If  the  above  interest  is  payable  in  gold,  at 
a  premium  of  40%,  what  would  be  the  current  value? 

47.  Sold  a  horse  and  carriage  and  lost  $50  by  the  trans- 
action. This  was  10%  of  the  cost  What  was  the  cost  and 
selling  price?         Ans.  $500.00,  cost;  $450.00,  selling  price. 

48.  If  $200  gain  $24  in  2  years,  what  will  $50  gain  in 
the  same  time  ? 

49.  Bought  15  shares  of  bank  stock  for  $1650,  and  sold 
them  at  an  advance  of  $5.50  a  share.  What  per  cent  did  I 
gain?  Ans.  5%. 

50.  Bought  160  acres  of  land  for  $15  an  acre.  Sold  10 
house-lots  at  $50  each,  a  quantity  of  lumber  and  wood  for 
$1725,  and  the  remainder  of  the  land  for  $1550.  What  did 
I  gain  ? 


THE      METEIC     SYSTEM 

or     WEiaHTS     AND      MEASURES. 

d03.  In  the  Metric  System  the  Scales  are  all  decimal 
as  in  United  States  Money.  It  is  so  named  from  the  Meter, 
which  is  one  ten-millionth  of  a  quadrant,  or  one  forty-millionth 
of  the  circumference  of  the  earth  measured  over  the  poles. 

Long  Measure. 

S04:.  The  principal  unit  of  length  is  the  Meter ^  which  is 
39.31  inches  long. 

i403 .    What  is  said  of  the  scales  of  the  Metric  System  ?    Why  is  this  System 
80  called  f 
iJ04.    Whatis  the  principal  unit  of  Long  Measure?     What  is  its  length? 


178 


THE    METRIC    SYSTEM. 


TABLE. 


10  Millimeters  (™'") 

make 

1  Centimeter. 

10  Centimeters 

1  Decimeter. 

10  Decimeters 

1  METEE  n. 

10  Meters 

1  Dekameter. 

10  Dekameters 

1   Hectometer. 

10  Hectometers 

1  Kilometer  C^'"). 

Centiin.                             Miu. 

Dccim. 

1=              10 

Meters. 

1  = 

10=            100 

Dekam.                  1  = 

10  = 

100=         1,000 

Hectom.              1  =           1 0  Z= 

100  = 

1,000=        10,000 

K„.               1=      10=       100  = 

1,000  = 

10,000=     100,000 

1       =10=100  =  1,000  = 

10,000  = 

100,000=1,000,000 

Note  1.  About  twenty-five  (more  exactly  25.4)  millimeters 
make  one  inch.  The  meter  is  about  three  feet,  three  inches,  and 
three-eighths  of  an  inch,  which  may  be  remembered  as  the  rule  of 
the  three  threes. 

Note  2.  The  kilometer  is  the  common  unit  for  road  measure' 
and  is  about  two  hundred  rods,  or  five-eighths  of  a  mile.  Five  meters 
make  about  one  rod. 

The  accompanying  scale  exhibits  one  decimeter  divided  into 
ten  centimeters,  each  centimeter  being  divided  into  ten  milli- 
meters. With  it  is  a  four-inch  scale  divided  into  eighths  of 
an  inch. 


Give  the  Table  of  Long  Measure.    What  is  the  Common  Unit  of  Road  Meas- 
ure ?    Its  length  ?    Draw  a  section  of  the  accompanying  scale,  and  explain  it. 


THE    :.IETRIC    SYSTEM. 


179 


These  measures,  as  well  as  all  the 
other  metric  measures  and  weights,  are 
written  like  whole  numbers  and  deci- 
mals. Thus,  3  kilometers,  8  hectome- 
ters, 7  meters,  and  5  decimeters,  are 
written  3807. 5°^.  Large  distances,  as 
in  road  measure,  are  given  as  kilome- 
ters and  decimals.  Thus,  47.34'"'" 
stands  for  4  myriameters,  7  kilometers, 
3  hectometers,  and  4  dekameters. 
Small  distances  are  usually  expressed 
in  millimeters,  or  in  centimeters. 

The  names  of  the  several  larger 
units  of  length  are  formed  from  the 
word  Meter,  by  prefixing  Myria  for 
10,000,  Kilo  for  1000,  Hecto  for  100, 
and  Deka  for  10.  The  smaller  units 
are  denoted  by  Deci  for  yV*  G^nti  for 
y^^,  and  Milli  for  -t^jVu-  I^  ^^  same 
way,  as  will  be  seen  hereafter,  are 
formed  the  names  of  weights  and  of 
measures  of  surface  and  capacity. 

Note  1.  The  first  series  of  prefixes  is 
from  the  Greek,  the  second  from  the  Latin 
language. 

Note  2.  The  terms  Dime,  Cent,  and 
Mill,  in  United  States  money,  for  the  tenth,  hundredth,  and  thou- 
sandth parts  of  a  dollar,  are  analogous  to  the  terms  Decimeter, 
Centimeter,  and  Millimeter. 

Note  3.     The  Metric  System  is  used  in  France  and  many  other 
countries,  and  is  legalized  in  the  United  States  and  Great  Britain. 

How  are  these  weights  and  measures  written  ?    Illustrate.    What  prefixes 
indicate  the  larger  denominations  ?    What  the  smaller  ? 


o 

Q 


CO 

M^ 

o          E 

H 

H             ::; 

2  o, —  I 

H                 - 
H            —I 

f                - 

<s> 

~  4 
-    p 

« — E 

w 

o 

* 1 

180  THE   METRIC    SYSTEM. 

^Oo.  To  reduce  ti  larger  denomination  in  the  Met- 
ric Syitem  to  a  smaller,  or  a  smaller  to  a  larger : 

Multiply  or  divide  hy  10,  100,  1000,  ^c,  as  the  case  may 
require.      {Art,    159.) 

Ex.  1.     Keduce  64  meters  to  millimeters. 

!■"  =  1000™™  64"  —  1000™"  X  64=  64000™™.  Ans. 

2.  Ecduce  8500  millimeters  to  meters.  Ads.  8.5™. 

3.  Keduce  95000  meters  to  kilometers.  Ans.  95^™. 
906.     Metric  measuies  and  weights  are  added,  subtracted, 

multiplied  and  divided  like  whole  numbers  and  decimals. 

Ex.  1.     Add  4.5™,  26.25™  and  9450™™.         Ans.  40.2™. 

2.     From  978™  take  392.64™.  Ans.  585.36™. 

iJ.     Multiply  736.45™  by  7. 
,     4.     Divide  1840.86™  by  63.  Ans.  29.22™. 

f  Square  Measure. 

d07.  The  principal  units  of  square  measure  are  the  Are 
and  the  Square  Meter.  The  Are  is  a  square  whose  side  is  10 
meters,  and  therefore  contains  100  square  meters. 

TABLE. 

100  Sq.  Centimeters  make  I  Sq.  Decimeter. 

100  Sq.  Decimeters  "  1  Centare,  or  sq.  meter. 

100  Centares,  or  sq.  meters    '*  1  Are  (*"■). 

100  Ares  '*  1  Hectare  C»). 

100  Hectares  '*  1  Sq.  Kilometer. 

Sq.  Decim.  Sq.  Centini. 

Sq.  Meters,                                 \ IQO 

or  Centurus.  — 

Ares.                        1  _                           100  —  10,000 

1  —  100  -  10,000  —  1,000,000 

8q.      1  _  100  —       10,000  =:         1,000,000  —         100,000,000 

1  =  100  -     10,000  —  1,000,000  —     100,000,000  —    10,000,000,000 

;i05.  How  is  Reduction  performed?  ^06.  How  are  these  measures  added, 
subtracted  &c. ?  307.  What  are  the  principal  units  of  square  measure? 
Give  the  Table. 


THE   METRIC    SYSTEM.  181 

Note  1.  The  hectare,  which  is  a  common  unit  for  land  measure, 
is  a  square  whose  side  is  a  hundred  meters ;  hence  it  is  equal  to 
10,000  square  meters.     It  is  2.471  acres. 

Notes  2.  Since  the  scale  in  square  measure  is  100  (two  dimen- 
sions, 10  X  10),  there  will  be  two  figures  for  each  denomination. 
Thus,  25  hectares,  7  ares,  17  centares,  and  20  square  decimeters, 
would  be  written  2507.172  ares,  or  250717.2  square  meters. 

Ex.  1.    Reduce  15  hectares  to  sq.  meters.    Ans.  150,000. 

2.  Reduce  456000  sq.  decimeters  to  ares.       Ans.  45.6. 

3.  Reduce  78  kilometers  to  ares. 

4.  Reduce  9624  ares  to  hectares.  Ans.  96.24. 

5.  In  a  field  300  meters  long  and  78  meters  wide,  how 
many  ares  ?  Ans.  234. 

6.  How  many  hectares  in  a  field  275  meters  long  and  500 
meters  wide?  Ans.  13.75. 

Cubic  Measure. 

208.  The  principal  unit  of  cubic  measure  is  the  Cubic 
Meter  or  Stere,     It  is  1.308  cubic  yards. 

TABLE. 

1000  Cub.  Centimeters     make     1  Cub.  Decimeter,  or  Z*Ver. 
1000  Cub.  Decimeters         "        1  Cub.  Meter,  or  Stere  C*). 
The  tenth  part  of  the  Stere  is  the  Decistere,  and  ten  Steres 
make  a  Dekastcre. 

Note.  Since  the  scale  is  a  thousand  (three  dimensions,  10  x 
10  X  10),  three  figures  will  be  required  for  each  denomination. 

207.  The  common  unit  for  land  measure  ?  Equal  to  how  may  acres  ? 
How  many  figures  required  for  each  denomination  ?    Why  ? 

308.  What  is  the  principal  unit  of  cubic  measure  ?  Equal  to  how  many 
cubic  yards  ?  Give  the  Table.  How  many  figures  required  for  each  denomina- 
tion?   Why? 


182 


THE    METRIC    SYSTEM. 


Ex.  1.     Reduce  12  cubic  meters  to  cubic  decimeters. 

Ans.  12000. 
Reduce  41,930,000  cubic  centimeters  to  cubic  meters  or 


2. 

stores. 
3. 
4. 
5. 
6. 


Ans.  41.93^*. 
Ans.  290. 


Reduce  29  steres  to  decisteres. 

Reduce  495  steres  to  dekasteres. 

Reduce  5230  decisteres  to  steres.  Ans.  523. 

In  a  pile  of  wood  10  meters  long,  1  meter  wide  and  3 
meters  high,  how  many  steres  ?  Ans.  30. 

7.     How  many  cubic  meters  in  a  box  3  meters  long,  1.25"" 
wide  and  1.2™  deep? 


Dry  and  Liquid  Measure. 

d09.  The  principal  unit,  both  for  Dry  and  for  Liquid 
Measure,  is  the  Cubic  Decimeter,  or  Liter,  It  is  a  little 
larger  than  a  wine  quart. 

TABLE. 

10  Milliliters,  or  cub.  centimeters,  mako  1    Centiliter  (°^). 

10  Centiliters  "  1  Deciliter. 

10  Deciliters  "  1  Liter  Q,  °'-<=»b<^«">"iter. 

10  Liters  "  1  Dekaliter. 

10  Dekaliters  "  1  Hectoliter  (^^). 

10  Hectoliters  "  1  Kiloliter,  o^cub.  meter. 


Centiliters.        Milliliters. 

Deciliters.                    1  ^= 

10 

1=          10  = 

100 

Delcaliters.               1==               10=              100  = 

1,000 

1=            10=            100=          1,000  = 

10,000 

Cnb. 
Meter. 

1=    10=     100=    1,000=    10,000  = 

100,000 

1   - 

-  10  =  100  =  1,000  =  10,000  =  100,000  = 

1,000,000 

309.    What  is  the  principal  unit  of  Dry  and  Liquid  Measure  ?    How  does 
it  compare  with  a  wine  quart  ?    Give  the  Table. 


THE   METRIC    SYSTEM.  183 

Note  1.  These  measures  are  usually  written  as  liters  and  decimal 
parts;  or  as  hectoliters  and  decimal  parts.  Thus,  2  kiloliters,  7 
hectoliters,  7  liters,  and  5  deciliters  are  written  27.075^,  or  2707.51. 

NoTB  2.  The  calculation  of  the  contents  of  a  bin  or  cistern,  is 
very  simple  in  the  metric  system.  The  product  of  the  length, 
breadth  and  thickness  in  decimeters,  gives  the  contents  in  liters. 

Ex.  1.     Eeduce  24000  milliliters  to  liters.  Ans,  24. 

2.  Eeduce  3560  liters  to  hectoliters.  Ans.  35.6. 

3.  Eeduce  .917  liters  to  centiliters. 

4.  Kow  many  liters  in  a  tank,  4.5™  long,  3™  wide,  and 
2.5""  deep  ?  45  X  30  X  25  =  33750.  Ans. 

5.  Eeduce  61250  liters  to  cubic  meters.       Ans.  61.25. 

6.  What  is  the  value  of  a  hectoliter  of  molasses  at  45  cts. 
a  liter?  Ans.  $45.00. 

Weight. 

dlO.  The  principal  units  of  weight  are  the  Gram  and 
the  Kilogram.  The  Kilogram  is  the  weight  of  a  liter  of 
water,  and  is  a  little  more  than  2-^  pounds  avoirdupois. 

TABLE. 

10  Milligrams  ('"^)  make  1  Centigram. 
10  Centigrams  "     1  Decigram. 

10  Decigrams  "     1  Gram  (^). 

10  Grams  *♦     1  Dekagram. 

10  Dekagrams  •*     1  Hectogram. 

10  Hectograms  "     1  Kilogram,  or  Kilo,  (^^), 

10  Kilograms  **     1  Myriagram. 

10  Myriagrams  "     1  Quintal. 

10  Quintals  "     1    TmineauQ), 

309.  How  are  these  measures  usually  written? 

310.  What  are  the  principal  units  of  weight?  What  is  a  kilogram? 
Equal  to  how  many  pounds  avoirdupois  ?    Give  the  Table. 


184 


THE   METRIC    SYSTEM, 


OentJgr. 

MilHgr. 

Deoigr. 

1= 

10 

Oiuni. 

1= 

10= 

100 

Dektgr. 

1= 

10= 

100= 

1000 

Heotogr.              1= 

10= 

100= 

1000= 

10,000 

Kilogr. 

1=            10= 

100= 

1000= 

10,000= 

100,000 

Myriagt.     1= 

10=        100= 

1000= 

10,000= 

100,000= 

1,000,000 

Quta.      1=     10= 

100=      1000= 

10,000= 

100,000= 

1,000,000— 

10,000,000 

,.  1—  10—  100— 

1000=  10,000= 

100,000=  1,000,000=  10,000,000= 

100,000,000 

1=10=100=1000=10,000=100,000=1,000,000=10,000,000=100,000,000=1,000,000,000 

Note.  A  cubic  centimeter  of  water  weighs  a  gram,  and  a  cubic 
meter  of  water  weighs  a  tonneau.  The  kilogram  is  often  called  kilo, 
for  brevity. 

Ex.  1.     Eeduce  64.73  kilos  to  grams.  Ans.  64730. 

2.  Eeduce  7490  kilos  to  tonneaus.  Ans.  7.49. 

3.  Eeduce  28500  milligrams  to  grams.  Ans.  28.6. 

4.  "What  is  the  weight  of  36  liters  of  water? 

Ans.  36=^^. 

5.  If  2^^^'  of  sugar  cost  80  cts.  what  will  be  the  expense 
of  le'^*^-   at  the  same  rate?  Ans.  $6.40 

6.  If  o5e  tonneau  of  coal  cost  $10,  what  will  7^  tonneaus 
cost? 


The  names  of  the  metric  weights  and  measures  are  formed  accord- 
ing to  a  simple  law,  as  will  be  seen  by  inspection  of  the  following 
scheme : 


Lengths. 


Surfaces.      Capacities.       Weights.       Eatios. 


Myria  -  meter.  Myria  -  gram. 

Kilo     -  meter.  Kilo    -  liter.  Kilo     -  gram. 

Hecto  -  meter.     Hect  -  are.     Hecto  -  liter.   Hecto  -  gram. 


Deka  -meter. 

Meter.  Are. 

Deci    -  meter. 
Centi  -  meter.    Cent  -  are. 
Milli  -meter. 


Deka  -  liter.   Deka  -  gram. 

Liter.  Gram. 

Deci  -  liter.  Deci  -  gram. 
Centi  -  liter.  Centi  -  gram. 
Milli   -  liter.   Milli    -  gram. 


10000 

1000 

100 

10 

1 

1 

TTnTTT 


THE   METRIC   SYSTEM. 


185 


311.  The  following  equivalents  of  the  metric  measures 
and  weights  have  been  established  by  Congress  for  use  in  all 
legal  proceedings : 


Measures    op   Length. 


METRIC  r)ENO> 
VAL 

nXATIONS  AND 
UES. 

EQUIVALENTS    IN    DENOMINATIONS 
IN  USE. 

Myriameter    .    . 

10,000  meters   .   . 

6.2137  miles. 

Kilometer  .    .    . 

1,000  meters  .    . 

0.62137  mile,  or  3,280  feet  and  10  in. 

Hectometer    .    . 

100  meters   .  . 

328  feet  and  1  inch. 

Dekameter     .     . 

10  meters   .  . 

393.7  inches. 

Meter     .... 

1  meter  .  .   . 

39.37  inches. 

Decimeter  .    .    . 

ytj.  of  a  meter 

3.937  inches. 

Centimeter      .    . 

^^^  of  a  meter 

0.3937  inch. 

Millimeter .    .    . 

TT)Vo-o^»™^*«'' 

0.0394  inch. 

Measukes  op  Surface. 


METRIC  DENOMINATIONS  AND 
VALUES. 


Hectare 
Are  .  , 
Centare 


10,000  eq.  meters 
100  sq.  meters 
1  eq.  meter 


EQUIVALENTS    IN   DENOMINATIONS 
IN  USE. 


2.471  acres. 

119.6  square  yards. 

1550  square  inches. 


Measures  op  Capacity. 


METRIC  DENOMINATIONS  AND  VALUES. 


Names. 


Kiloliter  or  stere 
Hectoliter 
Dekaliter . 
Liter  .  . 
Deciliter  . 
Centiliter. 
Milliliter . 


No.  of 
Liters. 


1000 

100 

10 

1 


1 
0 


1 
Tot> 


Cubic  Measure. 


1  cubic  meter 
^j^  of  a  cu.  meter 
10  cu.  decimeters 
1  cu.  decimeter 
-Ig-ofacu.  decim. 
10  cu.  centimeters 
1  cu.  centimeter 


EQUIVALENTS    IN  DENOMINA- 
TIONS IN  USE. 


Dry  Measure. 


Liquid  or  Wine 
Measure. 


1 .308  cu.  yards 

204.17  gallons. 

2bu.&3..35pks. 

26.417  gallons. 

9.08  quarts    .    . 

2.6417  gallons. 

0.908  quart    .    . 

1.0567  quarts. 

6.1022  cu.  inches 

0.845  giU. 

0.6102  cubic  inch 

0.338  fluid  ounce. 

0.061  cubic  inch 

0.27 fluid  drachm. 

186 


THE    METRIC   SYSTEM. 


Weights. 


METRIC  DENOMINATIONS  AND  VALUES. 


Names. 


Millier  or  tonneau 
Quintal  ,  .  . 
Myriaffram  .  . 
Kilogram  or  kilo 
Hectogram  .  . 
Dekagram  .  . 
Gram  .... 
Decigram  .  . 
Centigram  .  . 
Milligram    .    . 


No.  of 
Grams. 


1000000 

100000 

10000 

1000 

100 

10 

1 
tV 


Weight  of  what  quanti 
ty  of  water  at  maximum 
density. 


1  cubic  meter  .... 

1  hectoliter 

10  liters 

1  liter 

1  deciliter 

10  cubic  centimeters  .  . 
1  cubic  centimeter    .    . 
Y*^  of  a  cubic  centimeter 
10  cubic  millimeters  .  . 
1  cubic  millimeter    .    . 


EQUIVALENTS  IN  DE- 
NOMINATIONS   IN    USE. 


Avoirdupois  Weight. 

2204.6  pounds. 
220.40  pounds. 
22.046  pounds. 
2.2046  pounds. 
3.5274  ounces. 
0.3527  ounce. 
15.432  grains. 
1.5432  grains. 
0.1543  grain. 
0.0154  grain. 


212,    To   reduce   metric  weights    or   measures   to 
those  in  customary  use. 

Ex.  1.     Eeduce  2.5  kilos  to  pounds  avoirdupois. 

Ans.  55.115  lbs. 
Since  one  kilo  is  2.2046  pounds,  25  kilos  will  be  2.2046  X 
25  1=  55.115  lbs.     Hence,  we  have  the 

Etjle.     Multiply  the  number  of  metric  units  hy  the  corres- 
'ponding  numlter  in  the  table. 

2.  Reduce  30  meters  to  inches.  Ans.  1181.1. 

3.  Eeduce  18  liters  to  wine  quarts.  Ans.  19.0206. 

4.  Eeduce  50  hectares  to  acres.  Ans.  123.55. 


'■ZVif.    How  are  metric  weights  and  measures  reduced  to  those  in  custom- 
.ary  use? 


THE    METRIC    SYSTEM.  187 

313.  To  reduce  customary  weights  and  measures 
to  those  of  the  metric  system. 

Ex.  1.     Eeduce  195  inches  to  meters.  Ans.  4.953. 

Since  one  meter  is  39.37  inches,  the  numher  of  meters  in 
195  inches  is  the  numher  of  times  that  195  contains  39.37, 
that  is,  4.953  -|-  meters.     Hence,  we  have  the 

KuLE.  Divide  the  numher  of  the  customary  denomination 
by  the  corresponding  numher  in  the  table. 

2.  Eeduce  85  gallons  to  hectoliters.  Ans.  3.217. 

3.  Eeduce  28  ounces  avoirdupois  to  grams.  Ans.  7.938. 

4.  Eeduce  674  square  yards  to  ares.  Ans.  5.635. 


Miscellaneous   Examples. 

1. 

Add  65".  and  8000-™. 

Ans.  73-. 

2. 

Add  58.29^  and  136^ 

Ans.  6965^ 

3. 

Add  4».  60''^.  and  3620^. 

Ans.  4063.62'^^. 

4. 

Erom4'^^.  take  371^. 

Ans.  3629^. 

5. 

From  5^^  take  45^ 

Ans.  4558'. 

6. 

Multiply  24.5"^™.  hy  160. 

Ans.  3.92°», 

7. 

Multiply  42.35^.  by  40. 

Ans.  1694^. 

8. 

Divide  9^  hy  15^=^ 

Ans.  60. 

9. 

Divide  43.46'^'".  hy  106. 

Ans.  410™. 

10. 

Divide  126^.  hy  42^. 

Ans.  3. 

11. 

Eeduce  68.49^^*.  to  ares. 

Ans.  6849". 

12. 

Eeduce  lOt.  to  pounds  avoirdupois. 

Ans.  220461h. 

13. 

In  100  cords  of  wood,  how  many  { 

sterea  ? 

Ans.  362.4**. 
14.     What  cost  15m.  of  cloth  at  $3.00  a  meter  ? 

Ans.  $45.00. 

313     How  are  the   customary  measures  reduced  to  those  of  the  Metric 
System? 


188  THE   METRIC    SYSTEM. 

15.  How  many  kilograms  of  sugar  at  50  cts.  a  kilogram 
can  be  bought  for  $17.50  ? 

16.  How  many  kilometers  from  Boston  to  Albany,  the  dis- 
tance being  200  miles?"  Ans.  321.86''"-. 

17.  In  4  cubic  meters  of  water,  how  many  gallons? 

Ans.  1056.68. 

18.  What  is  the  weight  in  pounds  avoirdupois  of  2  cubic 
meters  of  water?  Ans.  4409.2. 

19.  If  a  hectoliter  of  corn  costs  $2.70,  what  is  the  price 
of  a  dekaliter?  Ans.  $27. 

20.  How  many  steres  will  a  pile  of  wood  contain  that  is 
20™  long,  2™  wide  and  3°»  high?  Ans.  120**. 

21.  Bought  500  hectares  of  land  at  $75.00  a  hectare  and 
sold  it  at  $100.00  a  hectare.  What  was  the  whole  gain,  and 
gain  per  cent?  Ans.  $12500,  and  33 J  percent. 

22.  A  meter  of  cloth  costs  $5.00  ;  how  many  yds,  can  be 
bought  for  $205.00  ?  Ans.  44.83 

23.  How  many  hectoliters  in  94  gallons?    Ans.  3.557. 

24.  What  is  the  cost  of  a  quintal  of  coffee  at  60  cents  a 
kilogram  ?  Ans.  $60.00. 

25.  Mount  Washington  is  6226  feet  above  the  level  of  the 
sea ;  what  is  its  height  in  meters  ? 

26.  The  difference  in  latitude,  between  New  Orleans  and 
Alton  Illinois,  is  9  degrees,  (one  tenth  of  a  quadrant.  Art. 
203, )  what  is  the  distance  in  kilometers  between  the  places  ? 

Ans.     lOOO'^™. 


VB   17363 


^^  1 


EATON'S  ABITHMETICS, 


THIS  SERIES  PRESI  ViS  A  FULL  AND  PRACTICAL  EXPOSITION  OF 

THE  METRIC   SYSTEM  OF   WEIGHTS   AND   MEASURES,  AND 

THE    LATENT   AND    AIOST   IMPROVED    METHODS 

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The  Primary  Arithmetic. 

Beautifully  illustrated,  and  made  very  attractive  for  beginners. 
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The  Elements  of  Arithmetic. 

A  short  cour^e  of  Written  Arithmetic,  with  a  large  immber  oi  ■ 
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The  Common  School  Arithmetic. 

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rules  ard  definitions,  and  is  full  Enough  for  all  ordinary  purpose^ 
It  is  especially  designed  for  Common  and  Grammar  Schools. 

The  Hi^h  8chooi  Arithmetic. 

A  thorough  and  exhaustive  treatise  for  High  Schools  and 
Academies. 

The  Grammar  ISchool  Arithmetic. 

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i  prominent  educators. 
I  TAGGARD  &  THOMPSON,  Publishers. 

I  29  CornhiU,  Boston. 


